Browse Geometry problems in the Studdy Learning Bank!

Problem 101

Rewrite $x^2 + y^2 - 18x - 2y + 1 = 0$ in standard form of a circle equation by completing the square.

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Problem 102

Find the perimeter and area of a parallelogram with sides of length $4.5$ feet and $9$ feet, and distance between $9$ -foot sides of $0.8$ foot.

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Problem 103

Find the value of $x$ in a triangle with angles $2x$ , $x$ , and $60$ degrees.

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Problem 104

Identify the point that represents the ordered pair $\left(5,-2\right)$ .

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Problem 105

Find the dimensions of a rectangle with upper-left coordinates $(-4,4)$ and upper-right coordinates $(4,4)$ that has a perimeter of 20 units.

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Problem 106

Given parallel lines $m$ and $n$ , find the relationship between the angles $(6x + 5)$ and $(5x - 12)$ and solve for $x$ .

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Problem 107

Identify the quadrant or axis for each coordinate pair: $(9,4)$ , $(0,-2)$ , $(6,0)$ , $(-4,-15)$ , $(-8,6)$ , $(3,-1)$ .

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Problem 108

Find the equation of the median from vertex $X$ for the triangle with vertices $X(1,-2), Y(7,4)$ , and $Z(8,-7)$ .

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Problem 109

Plot the points $B(-2,7), C(-9,4), D(-6,-3)$ , and $E(1,0)$ to form a quadrilateral BCDE, then graph it.

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Problem 110

Find the quadratic function for the area of a rectangular painting $29 \times 17$ inches with a frame of width $x$ on all sides.

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Problem 111

Find the length of $Y X$ if $Y$ is between $X$ and $Z$ , $Y Z=4$ , and $X Z=12$ .

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Problem 112

Plot the image of a triangle with vertices at $(-3,6)$ , $(6,0)$ , and $(-6,-9)$ dilated by a factor of $\frac{1}{3}$ centered at the origin.

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Problem 113

Find if $\triangle ABC$ with slopes $\frac{3}{4}$ and $-\frac{4}{3}$ is isosceles, right-angled, parallel, or equilateral.

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Problem 114

Find the side length of a square with a diagonal of $8\mathrm{~cm}$ .

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Problem 115

Find the surface area of a box with length $l=27$ feet, width $w=7$ feet, and height $h=9$ feet.

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Problem 116

Find the range of possible values for $x$ where the sides of a triangle have lengths $10x-6$ , $6x-15$ , and $x+24$ .

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Problem 117

Find the volume of a right cylinder with base defined by $y=x^{2}-6x$ and height 5.

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Problem 118

Find the dimensions of a rectangular playing field where the length is 8 yards longer than triple the width and the perimeter is 496 yards.
$\text{Length} = 3\times\text{Width} + 8$ $\text{Perimeter} = 2\times(\text{Length} + \text{Width}) = 496$

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Problem 119

Find the person with the greatest potential energy: $A)$ Bicycle rider, $B)$ Chair sitter, $C)$ Walker, $D)$ Slide-top sitter.

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Problem 120

Find the volume of a cube with side length $s$ , given the formula $V=s^{3}$ .

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Problem 121

Find the scale factor $k$ when $\text{EB}=12$ and $\text{EB'}=4$ . Solve for $x$ when $x+4=12$ .

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Problem 122

Find the center of the circle with equation $(x+3)^{2} + y^{2} = 4$ . Provide the simplified ordered pair.

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Problem 123

Find the value of $x$ in an isosceles trapezoid $VWXY$ where $VX=3x+85$ and $WY=4x+68$ .

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Problem 124

Find the length of $EF$ given that $\triangle ABC \cong \triangle DEF$ with $AB=2x-1$ , $BC=2x+5$ , and $DE=5x-4$ .

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Problem 125

Find the cosine ratio of $\angle \mathrm{L}$ in $\triangle \mathrm{KLM}$ with $\angle \mathrm{M}=90^{\circ}, \mathrm{ML}=9, \mathrm{KM}=40, \mathrm{LK}=41$ .

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Problem 126

A person walks at 2 m/s as they turn a corner. Does their $\mathbf{speed}$ , $\mathbf{velocity}$ , or $\mathbf{acceleration}$ change?

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Problem 127

Find the area of the polygon with vertices at $W(2,10), X(4,10), Y(4,4)$ , and $Z(2,4)$ . (square units)

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Problem 128

Calculate the number of wheel rotations needed to travel 200 feet, given a wheel diameter of 2.33 feet and $\pi = 3.14$ .

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Problem 129

Find the maximum area of a rectangle formed by bending a $68 \mathrm{~cm}$ wire. Write a formula connecting the area $A$ and the length $x$ , then sketch the graph of $A$ against $x$ to find the maximum area.

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Problem 130

Determine the area of the remaining part of a $20 \, \text{ft} \times 15 \, \text{ft}$ flag after a $10 \, \text{ft} \times 14 \, \text{ft}$ triangle is cut from the center.

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Problem 131

Find $x$ given $AC=22$ , $BC=x+14$ , and $AB=x+10$ .

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Problem 132

Find the length of line segment AC if AB=16 and BC=12, given that points A, B, and C are collinear with B between A and C.

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Problem 133

Find the value of $h$ when the area $A=38.64$ and the base lengths are $a=5.5$ and $b=3.7$ using the formula $A=\frac{(a+b) h}{2}$ .

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Problem 134

Find the value of $x$ such that the line segment $\overline{NO}$ is parallel to $\overline{PQ}$ , given $N(-1,-1), O(9,9), P(-8,3)$ , and $Q(x, 2)$ .

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Problem 135

Find the ratio of green (0.8 m), blue (1.6 m), and red (0.6 m) paint segments on a pole. Express the ratio in simplest form using whole numbers.

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Problem 136

Find the distance between two points $A(2,-3)$ and $B(4,5)$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

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Problem 137

Find the mathematical relationship between the points $(8,6)$ and $(3,-6)$ , such as the distance or slope of the line.

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Problem 138

1. Formulate and rearrange formulas: a) A rectangle has a perimeter of $28 \mathrm{~cm}$ . Express its area in terms of a side length $x$ . b) For a rectangle, one side $x$ is twice the other side $y$ . Express the area in terms of $x$ . c) For a cube, express the volume $V$ in terms of the total edge length $K$ , and vice versa. d) For a cube, express the volume $V$ in terms of the surface area $O$ , and vice versa. e) For a cuboid with sides $a, b, c$ where $a=2b$ and $b=2c$ , provide a formula for the volume in terms of each variable.

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Problem 139

Find the coordinates of point $T$ if the midpoint of line $RT$ is $(3,0)$ and the coordinates of $R$ are $(-1,-4)$ .

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Problem 140

Find the difference between the perimeters $P_1$ and $P_2$ of two figures, where the difference is $n$ . Write and solve an absolute value inequality to represent the situation.

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Problem 141

Gegeben ist eine Bushaltestelle mit quadratischem Dach, dessen Funktion $f(x)=-\frac{2}{9}x^2+0.5$ ist. Die Breite ist 3 m, Höhe ohne Dach 2 m, mit Dach 2.5 m, Tiefe 1 m. Bestimmen Sie die Koeffizienten des Dachprofils und das Volumen des Häuschens.

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Problem 142

Find the area $A(x)$ of a rectangular field with $160$ meters of fencing, where a side length is $x$ meters. Determine the side length $x$ that maximizes the area and the maximum area.
(a) $A(x) = x(160 - 2x)$ (b) The side length $x$ that maximizes the area is $x = 40$ meters. (c) The maximum area is $A(40) = 3200$ square meters.

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Problem 143

Use a grid to find the product of $1 \frac{1}{4}$ and $3 \frac{1}{4}$ , where each square represents $\frac{1}{4}$ ft by $\frac{1}{4}$ ft.

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Problem 144

Determine which triangle congruence proofs are valid: ASA, AAA, $\mathrm{HL}$ , AAS, SSA, SAS, CPCTC, SSS.

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Problem 145

Find the coordinates of point A given that its x-coordinate is 4 and its y-coordinate satisfies the equation $y=x-5$ .

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Problem 146

Find the quadrant of the angle $177^{\circ}$ .

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Problem 147

Find the speed of the tip of a woman's shadow when she is $15 \mathrm{~m}$ from a $3 \mathrm{~m}$ tall pole, running at $1.8 \mathrm{~m} / \mathrm{s}$ and is $1.5 \mathrm{~m}$ tall.

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Problem 148

Determine if the point $(2,-3)$ lies on, inside, or outside the circle defined by the equation $x^{2}+y^{2}=9$ .

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Problem 149

Identify the set of 3 numbers that could represent the sides of a triangle. Options: $\{4,16,21\}$ , $\{9,14,20\}$ , $\{11,14,27\}$ , $\{6,14,20\}$ .

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Problem 150

A figure has a perimeter of $16$ units. What is the perimeter after a dilation with a scale factor of $8$ ?

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Problem 151

Find the length of the longest diagonal in a right rectangular prism with dimensions 12.6 cm x 3.2 cm x 6 cm. The longest diagonal is $\sqrt{12.6^2 + 3.2^2 + 6^2}$ cm (rounded to the nearest tenth).

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Problem 152

Find the length of a pool with width 6 feet and perimeter $P=30$ feet using the formula $P=2L+2W$ .

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Problem 153

Find the radius $x$ of a cone with height $12 \mathrm{cm}$ and volume $452.16 \mathrm{cm}^{3}$ , using $\pi = 3.14$ .

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Problem 154

Find the length of the midsegment $\overline{VY}$ of a trapezoid UWXZ, given that $WX=3$ and $UZ=5$ .

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Problem 155

Calculate the volume of a cylinder with a 4-inch radius and 1-inch height, using $\pi = 3.14$ . Round the answer to the nearest hundredth.

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Problem 156

Plot the points $A(0,4), B(4,4), C(4,-4), D(0,-4)$ and find the perimeter of the resulting figure.

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Problem 157

Represent the set $(A \cap C) - B$ using a Venn diagram. Then graph the set using the provided graphing tool.

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Problem 158

Given: $B$ is the midpoint of $\overline{A C}$ , $\overline{A B} \cong \overline{C D}$ . Prove: $C$ is the midpoint of $\overline{B D}$ .

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Problem 159

Find the total surface area of a cone with radius $4 \mathrm{~cm}$ and slant height $6 \mathrm{~cm}$ using the formula $\pi r l$ , where $r$ is the radius and $l$ is the slant height. Give the answer in terms of $\pi$ .

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Problem 160

Identify which numbers 0-9 have rotational symmetry.

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Problem 161

Determine the equation of the lines passing through the points A(2/1), B(6/1), and C(4/5) that form a triangle.

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Problem 162

Find the pool area given capacity $75.6 \mathrm{m}^{3}$ and depth $120 \mathrm{cm}$ . Tape the circular jacuzzi circumference to nearest cm. Calculate the total floor area to be tiled, and the number of tile boxes needed.

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Problem 163

Find the surface area formula for a cylinder given its radius $r$ and height $h$ . Rewrite the formula in terms of diameter $d$ and identify the relevant part to calculate the surface area of the cylinder.

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Problem 164

Finde die Seitenlängen eines Rechtecks mit Fläche $400 \mathrm{~m}^{2}$ , sodass der Umfang minimal wird.

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Problem 165

Dibuje una línea que represente el "ascenso" y otra línea que represente el "recorrido". Indique la pendiente $m$ de la recta.

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Problem 166

Find the number of packages of 24 one-foot-square tiles needed to cover a $68 \times 68$ foot square ballroom floor, where partial packages cannot be bought.

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Problem 167

Calculate the volume of a hemisphere with a 60.9-inch diameter, rounded to the nearest 0.1 cubic inch.
$V = \frac{2}{3}\pi r^3$ , where $r = 60.9/2 = 30.45$ inches.

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Problem 168

Find the area of an isosceles right triangle with leg length $18 \mathrm{cm}$ given that a triangle with leg $12 \mathrm{cm}$ has area $72 \mathrm{sq} \mathrm{cm}$ and the area varies directly as the square of the leg length.

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Problem 169

Identify the center and radius of a circle with the equation $(x-4)^{2}+(y-1)^{2}=9$ , then graph the circle.

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Problem 170

Ant running at constant speed in circular path on clock face. Starts at 12 o'clock, moving clockwise. What is the direction of the ant's acceleration vector?

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Problem 171

Find $x$ given $AB = 7x-20$ , $BC = 4$ , and $AC = 3x$ .

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Problem 172

Find the straight-line distance from the starting point after driving 14 miles east and 10 miles north. Round to the nearest tenth of a mile.
Solution: The straight-line distance is given by $\sqrt{14^2 + 10^2} \approx 17.2$ miles.

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Problem 173

Find the value of $x$ given a rectangle $TUVW$ where $WX=4$ and $VX=-5x+4$ .

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Problem 174

Determine if the statement "A square is a parallelogram" is always, sometimes, or never true.

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Problem 175

Find the distance between the given points using the distance formula. Round to the nearest hundredth. 9) $(0,3)$ and $(4,-9)$ : $\sqrt{(4-0)^2 + (-9-3)^2} \approx 12.04$ 12) $(-1,5)$ and $(4,5)$ : $\sqrt{(4-(-1))^2 + (5-5)^2} = 5$ 10) $(-6,16)$ and $(11,-2)$ : $\sqrt{(11-(-6))^2 + (-2-16)^2} \approx 19.08$ 13) $(14,2)$ and $(-2,-2)$ : $\sqrt{(14-(-2))^2 + (2-(-2))^2} \approx 16.12$ 11) $(-8,7)$ and $(10,4)$ : $\sqrt{(10-(-8))^2 + (4-7)^2} \approx 18.03$ 14) $(-5,6)$ and $(-5,7)$ : $\sqrt{(-5-(-5))^2 + (7-6)^2} = 1$

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Problem 176

Find $x, y, z$ in a rhombus with equal sides, where one side is 8 units long.

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Problem 177

Graph the equation $x=9$ by plotting points. Determine the domain and range.

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Problem 178

Find the area of a scalene triangle with vertices at (2,7), (5,-6), and (8,-6).

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Problem 179

Find the height of a stack of DVD cases given the proportional relationship between height and number of cases. Determine the equation relating height $y$ and number of cases $x$ , then calculate the height of 13 DVD cases.

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Problem 180

Find the arc length of a circle with radius $r=10$ feet and central angle $\theta=320^\circ$ . Express the arc length $s$ in terms of $\pi$ , rounded to two decimal places.

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Problem 181

Find the value of $x$ in the parallelogram $QRST$ given that $QU=18$ and $US=-x+1$ .

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Problem 182

Draw a square with 10 equal columns. What is the decimal value of each column if the side length is 10?

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Problem 183

Graph the reflection of point $M(-7,-9)$ over the line $y=-x$ .

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Problem 184

Find the perimeter of the rectangle with vertices at $(6,-2)$ , $(8,-2)$ , $(6,3)$ , and $(8,3)$ .

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Problem 185

Find the directrix of the parabola $x = \frac{1}{24} y^2$ .

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Problem 186

Find the minimum cost to build a rectangular fence of 180 square yards, where one side uses $\$ 9$ per yard material and the other three sides use $\$ 1$ per yard material.

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Problem 187

Find Cartesian equation of curve with $r^{2}=15$ . Identify the curve: hyperbola, ellipse, limaçon, line, or circle.

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Problem 188

Find the equation of the ellipse with foci at $(0,-7)$ and $(0,7)$ and $x$ -intercepts at $-6$ and $6$ .

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Problem 189

Approximate the parabolic shape of the Humber River Pedestrian Bridge using the equation $h=-\frac{1}{144} x^{2}+\frac{5}{6} x$ , where $x$ is the horizontal distance in meters and $h$ is the height in meters. Graph the quadratic relation using a table of values.

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Problem 190

Graph a parallelogram VWXY with vertices V(-3,14), W(-6,11), X(-6,5), Y(-3,8) after a glide reflection: translate 18 units down, then reflect across $x=4$ .

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Problem 191

Find the distance between opposite corners of a $5 \times 7$ inch windowpane. Round to the nearest tenth of an inch.

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Problem 192

Find the final coordinates after moving left 2 units and down 1 unit from the starting point $(4,2)$ .

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Problem 193

Find the coordinates of the vertex $A'$ after rotating triangle $ABC$ with vertices $A(1,5), B(5,2)$ , and $C(-1,2)$ by $90^{\circ}$ .

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Problem 194

Find the length of AC given that AB=7 ft, BD=2 ft, and DC=4.5 ft. Round the final answer to 2 decimal places.

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Problem 195

Find the grid width and vertical axis scale to best plot the distance data over time. The data shows the distance (km) at various time points (minutes).

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Problem 196

Finde die Länge der Schenkel eines gleichschenkligen Dreiecks, wenn Basis $a=37$ cm und Höhe $h_c=35$ cm sind.

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Problem 197

A sailboat's shadow $\triangle ABC$ changes to $\triangle A'B'C'$ as the distance to the screen varies. Choose all that apply: it changed shape, size, vertices moved along rays, or position.

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Problem 198

Find the distance between the line $y = x - 4$ and the point $(-1, 5)$ . Round the answer to the nearest tenth.

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Problem 199

Find the formula for the area of a rectangle from the given options.
$A = L * W$

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Problem 200

Graph a triangle with vertices $A(3,1), B(3,4), C(7,1)$ . Classify the triangle by its sides and determine if it is a right triangle.

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Geometry

Problem 101

Rewrite x2+y2−18x−2y+1=0x^2 + y^2 - 18x - 2y + 1 = 0x2+y2−18x−2y+1=0 in standard form of a circle equation by completing the square.

Problem 102

Find the perimeter and area of a parallelogram with sides of length 4.54.54.5 feet and 999 feet, and distance between 999-foot sides of 0.80.80.8 foot.

Problem 103

Find the value of xxx in a triangle with angles 2x2x2x, xxx, and 606060 degrees.

Problem 104

Identify the point that represents the ordered pair (5,−2)\left(5,-2\right)(5,−2).

Problem 105

Find the dimensions of a rectangle with upper-left coordinates (−4,4)(-4,4)(−4,4) and upper-right coordinates (4,4)(4,4)(4,4) that has a perimeter of 20 units.

Problem 106

Given parallel lines mmm and nnn, find the relationship between the angles (6x+5)(6x + 5)(6x+5) and (5x−12)(5x - 12)(5x−12) and solve for xxx.

Problem 107

Identify the quadrant or axis for each coordinate pair: (9,4)(9,4)(9,4), (0,−2)(0,-2)(0,−2), (6,0)(6,0)(6,0), (−4,−15)(-4,-15)(−4,−15), (−8,6)(-8,6)(−8,6), (3,−1)(3,-1)(3,−1).

Problem 108

Find the equation of the median from vertex XXX for the triangle with vertices X(1,−2),Y(7,4)X(1,-2), Y(7,4)X(1,−2),Y(7,4), and Z(8,−7)Z(8,-7)Z(8,−7).

Problem 109

Plot the points B(−2,7),C(−9,4),D(−6,−3)B(-2,7), C(-9,4), D(-6,-3)B(−2,7),C(−9,4),D(−6,−3), and E(1,0)E(1,0)E(1,0) to form a quadrilateral BCDE, then graph it.

Problem 110

Find the quadratic function for the area of a rectangular painting 29×1729 \times 1729×17 inches with a frame of width xxx on all sides.

Problem 111

Find the length of YXY XYX if YYY is between XXX and ZZZ, YZ=4Y Z=4YZ=4, and XZ=12X Z=12XZ=12.

Problem 112

Plot the image of a triangle with vertices at (−3,6)(-3,6)(−3,6), (6,0)(6,0)(6,0), and (−6,−9)(-6,-9)(−6,−9) dilated by a factor of 13\frac{1}{3}31​ centered at the origin.

Problem 113

Find if △ABC\triangle ABC△ABC with slopes 34\frac{3}{4}43​ and −43-\frac{4}{3}−34​ is isosceles, right-angled, parallel, or equilateral.

Problem 114

Find the side length of a square with a diagonal of 8 cm8\mathrm{~cm}8 cm.

Problem 115

Find the surface area of a box with length l=27l=27l=27 feet, width w=7w=7w=7 feet, and height h=9h=9h=9 feet.

Problem 116

Find the range of possible values for xxx where the sides of a triangle have lengths 10x−610x-610x−6, 6x−156x-156x−15, and x+24x+24x+24.

Problem 117

Find the volume of a right cylinder with base defined by y=x2−6xy=x^{2}-6xy=x2−6x and height 5.

Problem 118

Problem 119

Find the person with the greatest potential energy: A)A)A) Bicycle rider, B)B)B) Chair sitter, C)C)C) Walker, D)D)D) Slide-top sitter.

Problem 120

Find the volume of a cube with side length sss, given the formula V=s3V=s^{3}V=s3.

Problem 121

Find the scale factor kkk when EB=12\text{EB}=12EB=12 and EB’=4\text{EB'}=4EB’=4. Solve for xxx when x+4=12x+4=12x+4=12.

Problem 122

Find the center of the circle with equation (x+3)2+y2=4(x+3)^{2} + y^{2} = 4(x+3)2+y2=4. Provide the simplified ordered pair.

Problem 123

Find the value of xxx in an isosceles trapezoid VWXYVWXYVWXY where VX=3x+85VX=3x+85VX=3x+85 and WY=4x+68WY=4x+68WY=4x+68.

Problem 124

Find the length of EFEFEF given that △ABC≅△DEF\triangle ABC \cong \triangle DEF△ABC≅△DEF with AB=2x−1AB=2x-1AB=2x−1, BC=2x+5BC=2x+5BC=2x+5, and DE=5x−4DE=5x-4DE=5x−4.

Problem 125

Find the cosine ratio of ∠L\angle \mathrm{L}∠L in △KLM\triangle \mathrm{KLM}△KLM with ∠M=90∘,ML=9,KM=40,LK=41\angle \mathrm{M}=90^{\circ}, \mathrm{ML}=9, \mathrm{KM}=40, \mathrm{LK}=41∠M=90∘,ML=9,KM=40,LK=41.

Problem 126

A person walks at 2 m/s as they turn a corner. Does their speed\mathbf{speed}speed, velocity\mathbf{velocity}velocity, or acceleration\mathbf{acceleration}acceleration change?

Problem 127

Find the area of the polygon with vertices at W(2,10),X(4,10),Y(4,4)W(2,10), X(4,10), Y(4,4)W(2,10),X(4,10),Y(4,4), and Z(2,4)Z(2,4)Z(2,4). (square units)

Problem 128

Calculate the number of wheel rotations needed to travel 200 feet, given a wheel diameter of 2.33 feet and π=3.14\pi = 3.14π=3.14.

Problem 129

Find the maximum area of a rectangle formed by bending a 68 cm68 \mathrm{~cm}68 cm wire. Write a formula connecting the area AAA and the length xxx, then sketch the graph of AAA against xxx to find the maximum area.

Problem 130

Determine the area of the remaining part of a 20 ft×15 ft20 \, \text{ft} \times 15 \, \text{ft}20ft×15ft flag after a 10 ft×14 ft10 \, \text{ft} \times 14 \, \text{ft}10ft×14ft triangle is cut from the center.

Problem 131

Find xxx given AC=22AC=22AC=22, BC=x+14BC=x+14BC=x+14, and AB=x+10AB=x+10AB=x+10.

Problem 132

Find the length of line segment AC if AB=16 and BC=12, given that points A, B, and C are collinear with B between A and C.

Problem 133

Find the value of hhh when the area A=38.64A=38.64A=38.64 and the base lengths are a=5.5a=5.5a=5.5 and b=3.7b=3.7b=3.7 using the formula A=(a+b)h2A=\frac{(a+b) h}{2}A=2(a+b)h​.

Problem 134

Find the value of xxx such that the line segment NO‾\overline{NO}NO is parallel to PQ‾\overline{PQ}PQ​, given N(−1,−1),O(9,9),P(−8,3)N(-1,-1), O(9,9), P(-8,3)N(−1,−1),O(9,9),P(−8,3), and Q(x,2)Q(x, 2)Q(x,2).

Problem 135

Find the ratio of green (0.8 m), blue (1.6 m), and red (0.6 m) paint segments on a pole. Express the ratio in simplest form using whole numbers.

Problem 136

Find the distance between two points A(2,−3)A(2,-3)A(2,−3) and B(4,5)B(4,5)B(4,5) using the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​.

Problem 137

Find the mathematical relationship between the points (8,6)(8,6)(8,6) and (3,−6)(3,-6)(3,−6), such as the distance or slope of the line.

Problem 138

Problem 139

Find the coordinates of point TTT if the midpoint of line RTRTRT is (3,0)(3,0)(3,0) and the coordinates of RRR are (−1,−4)(-1,-4)(−1,−4).

Problem 140

Find the difference between the perimeters P1P_1P1​ and P2P_2P2​ of two figures, where the difference is nnn. Write and solve an absolute value inequality to represent the situation.

Problem 141

Rewrite $x^2 + y^2 - 18x - 2y + 1 = 0$ in standard form of a circle equation by completing the square.

Find the perimeter and area of a parallelogram with sides of length $4.5$ feet and $9$ feet, and distance between $9$ -foot sides of $0.8$ foot.

Find the value of $x$ in a triangle with angles $2x$ , $x$ , and $60$ degrees.

Identify the point that represents the ordered pair $\left(5,-2\right)$ .

Find the dimensions of a rectangle with upper-left coordinates $(-4,4)$ and upper-right coordinates $(4,4)$ that has a perimeter of 20 units.

Given parallel lines $m$ and $n$ , find the relationship between the angles $(6x + 5)$ and $(5x - 12)$ and solve for $x$ .

Identify the quadrant or axis for each coordinate pair: $(9,4)$ , $(0,-2)$ , $(6,0)$ , $(-4,-15)$ , $(-8,6)$ , $(3,-1)$ .

Find the equation of the median from vertex $X$ for the triangle with vertices $X(1,-2), Y(7,4)$ , and $Z(8,-7)$ .

Plot the points $B(-2,7), C(-9,4), D(-6,-3)$ , and $E(1,0)$ to form a quadrilateral BCDE, then graph it.

Find the quadratic function for the area of a rectangular painting $29 \times 17$ inches with a frame of width $x$ on all sides.

Find the length of $Y X$ if $Y$ is between $X$ and $Z$ , $Y Z=4$ , and $X Z=12$ .

Plot the image of a triangle with vertices at $(-3,6)$ , $(6,0)$ , and $(-6,-9)$ dilated by a factor of $\frac{1}{3}$ centered at the origin.

Find if $\triangle ABC$ with slopes $\frac{3}{4}$ and $-\frac{4}{3}$ is isosceles, right-angled, parallel, or equilateral.

Find the side length of a square with a diagonal of $8\mathrm{~cm}$ .

Find the surface area of a box with length $l=27$ feet, width $w=7$ feet, and height $h=9$ feet.

Find the range of possible values for $x$ where the sides of a triangle have lengths $10x-6$ , $6x-15$ , and $x+24$ .

Find the volume of a right cylinder with base defined by $y=x^{2}-6x$ and height 5.

Find the person with the greatest potential energy: $A)$ Bicycle rider, $B)$ Chair sitter, $C)$ Walker, $D)$ Slide-top sitter.

Find the volume of a cube with side length $s$ , given the formula $V=s^{3}$ .

Find the scale factor $k$ when $\text{EB}=12$ and $\text{EB'}=4$ . Solve for $x$ when $x+4=12$ .

Find the center of the circle with equation $(x+3)^{2} + y^{2} = 4$ . Provide the simplified ordered pair.

Find the value of $x$ in an isosceles trapezoid $VWXY$ where $VX=3x+85$ and $WY=4x+68$ .

Find the length of $EF$ given that $\triangle ABC \cong \triangle DEF$ with $AB=2x-1$ , $BC=2x+5$ , and $DE=5x-4$ .

Find the cosine ratio of $\angle \mathrm{L}$ in $\triangle \mathrm{KLM}$ with $\angle \mathrm{M}=90^{\circ}, \mathrm{ML}=9, \mathrm{KM}=40, \mathrm{LK}=41$ .

A person walks at 2 m/s as they turn a corner. Does their $\mathbf{speed}$ , $\mathbf{velocity}$ , or $\mathbf{acceleration}$ change?

Find the area of the polygon with vertices at $W(2,10), X(4,10), Y(4,4)$ , and $Z(2,4)$ . (square units)

Calculate the number of wheel rotations needed to travel 200 feet, given a wheel diameter of 2.33 feet and $\pi = 3.14$ .

Find the maximum area of a rectangle formed by bending a $68 \mathrm{~cm}$ wire. Write a formula connecting the area $A$ and the length $x$ , then sketch the graph of $A$ against $x$ to find the maximum area.

Determine the area of the remaining part of a $20 \, \text{ft} \times 15 \, \text{ft}$ flag after a $10 \, \text{ft} \times 14 \, \text{ft}$ triangle is cut from the center.

Find $x$ given $AC=22$ , $BC=x+14$ , and $AB=x+10$ .

Find the value of $h$ when the area $A=38.64$ and the base lengths are $a=5.5$ and $b=3.7$ using the formula $A=\frac{(a+b) h}{2}$ .

Find the value of $x$ such that the line segment $\overline{NO}$ is parallel to $\overline{PQ}$ , given $N(-1,-1), O(9,9), P(-8,3)$ , and $Q(x, 2)$ .

Find the distance between two points $A(2,-3)$ and $B(4,5)$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Find the mathematical relationship between the points $(8,6)$ and $(3,-6)$ , such as the distance or slope of the line.

Find the coordinates of point $T$ if the midpoint of line $RT$ is $(3,0)$ and the coordinates of $R$ are $(-1,-4)$ .

Find the difference between the perimeters $P_1$ and $P_2$ of two figures, where the difference is $n$ . Write and solve an absolute value inequality to represent the situation.

Gegeben ist eine Bushaltestelle mit quadratischem Dach, dessen Funktion $f(x)=-\frac{2}{9}x^2+0.5$ ist. Die Breite ist 3 m, Höhe ohne Dach 2 m, mit Dach 2.5 m, Tiefe 1 m. Bestimmen Sie die Koeffizienten des Dachprofils und das Volumen des Häuschens.

Use a grid to find the product of $1 \frac{1}{4}$ and $3 \frac{1}{4}$ , where each square represents $\frac{1}{4}$ ft by $\frac{1}{4}$ ft.

Determine which triangle congruence proofs are valid: ASA, AAA, $\mathrm{HL}$ , AAS, SSA, SAS, CPCTC, SSS.

Find the coordinates of point A given that its x-coordinate is 4 and its y-coordinate satisfies the equation $y=x-5$ .

Find the quadrant of the angle $177^{\circ}$ .

Find the speed of the tip of a woman's shadow when she is $15 \mathrm{~m}$ from a $3 \mathrm{~m}$ tall pole, running at $1.8 \mathrm{~m} / \mathrm{s}$ and is $1.5 \mathrm{~m}$ tall.

Determine if the point $(2,-3)$ lies on, inside, or outside the circle defined by the equation $x^{2}+y^{2}=9$ .

Identify the set of 3 numbers that could represent the sides of a triangle. Options: $\{4,16,21\}$ , $\{9,14,20\}$ , $\{11,14,27\}$ , $\{6,14,20\}$ .

A figure has a perimeter of $16$ units. What is the perimeter after a dilation with a scale factor of $8$ ?

Find the length of the longest diagonal in a right rectangular prism with dimensions 12.6 cm x 3.2 cm x 6 cm. The longest diagonal is $\sqrt{12.6^2 + 3.2^2 + 6^2}$ cm (rounded to the nearest tenth).

Find the length of a pool with width 6 feet and perimeter $P=30$ feet using the formula $P=2L+2W$ .

Find the radius $x$ of a cone with height $12 \mathrm{cm}$ and volume $452.16 \mathrm{cm}^{3}$ , using $\pi = 3.14$ .

Find the length of the midsegment $\overline{VY}$ of a trapezoid UWXZ, given that $WX=3$ and $UZ=5$ .

Calculate the volume of a cylinder with a 4-inch radius and 1-inch height, using $\pi = 3.14$ . Round the answer to the nearest hundredth.

Plot the points $A(0,4), B(4,4), C(4,-4), D(0,-4)$ and find the perimeter of the resulting figure.

Represent the set $(A \cap C) - B$ using a Venn diagram. Then graph the set using the provided graphing tool.

Given: $B$ is the midpoint of $\overline{A C}$ , $\overline{A B} \cong \overline{C D}$ . Prove: $C$ is the midpoint of $\overline{B D}$ .

Find the total surface area of a cone with radius $4 \mathrm{~cm}$ and slant height $6 \mathrm{~cm}$ using the formula $\pi r l$ , where $r$ is the radius and $l$ is the slant height. Give the answer in terms of $\pi$ .

Find the pool area given capacity $75.6 \mathrm{m}^{3}$ and depth $120 \mathrm{cm}$ . Tape the circular jacuzzi circumference to nearest cm. Calculate the total floor area to be tiled, and the number of tile boxes needed.

Find the surface area formula for a cylinder given its radius $r$ and height $h$ . Rewrite the formula in terms of diameter $d$ and identify the relevant part to calculate the surface area of the cylinder.

Finde die Seitenlängen eines Rechtecks mit Fläche $400 \mathrm{~m}^{2}$ , sodass der Umfang minimal wird.

Dibuje una línea que represente el "ascenso" y otra línea que represente el "recorrido". Indique la pendiente $m$ de la recta.

Find the number of packages of 24 one-foot-square tiles needed to cover a $68 \times 68$ foot square ballroom floor, where partial packages cannot be bought.

Calculate the volume of a hemisphere with a 60.9-inch diameter, rounded to the nearest 0.1 cubic inch.
$V = \frac{2}{3}\pi r^3$ , where $r = 60.9/2 = 30.45$ inches.

Find the area of an isosceles right triangle with leg length $18 \mathrm{cm}$ given that a triangle with leg $12 \mathrm{cm}$ has area $72 \mathrm{sq} \mathrm{cm}$ and the area varies directly as the square of the leg length.

Identify the center and radius of a circle with the equation $(x-4)^{2}+(y-1)^{2}=9$ , then graph the circle.

Find $x$ given $AB = 7x-20$ , $BC = 4$ , and $AC = 3x$ .

Find the straight-line distance from the starting point after driving 14 miles east and 10 miles north. Round to the nearest tenth of a mile.
Solution: The straight-line distance is given by $\sqrt{14^2 + 10^2} \approx 17.2$ miles.

Find the value of $x$ given a rectangle $TUVW$ where $WX=4$ and $VX=-5x+4$ .