13 Diberi segi tiga ABC dengan AB=4∼i−6∼j dan AC=2∼i+4∼j.T berada pada garis BC dengan keadaan 3BT=TC.
It is given that a triangle ABC with AB=4∼i−6∼j∼ndAC=2∼i+4∼j. T lies on the line BC such that 3BT=TC.
(a) Cari vektor Find the vector
(i) BC,
(ii) AT. Seterusnya, car1 vektor unit dalam arah AT.
AT. Hence, find the unit vector in the direction of AT.
[5 markah]
[5 marks]
(b) Jika D ialah satu titik dengan keadaan AD=hBC dan AD=−3∼i+∼kj, dengan keadaan h dan k adalah pemalar. Cari nilai h dan nilai k.
If D is a point such that AD=hBC and AD=−3i∼i+k, such that h and k are constants. Find the value of h and of k.
A rope is tied halfway to the top of an 8 m tree. The other end is tied to the ground 1 m from the tree. Which equation represents the rope's height y at distance x from the tree?
A. x−4y=4
B. 4x−y=4
C. 4x+y=4
D. x−4y=−4
Are the following statements true or false? False 1. For any scalar c and any vector v, we have ∥cv∥=c∥v∥. False 2. If v and w are any two vectors, then ∥v+w∥=∥v∥+∥w∥. False 3. (i×j)⋅k=i⋅(j×k). True 4. The value of v⋅(v×w) is always zero.
Which equation models a line that is perpedicular to y=−43x+1, and passes through the point (12,9) ?
A. y=34x−7
B. y=−34x+25
C. y−12=34(x−9)
D. y−12=−34(x−9)
Let P=(0,0,0),Q=(1,−1,−1),R=(−2,1,1).
Find the area of the triangle PQR.
area =□
Let T=(4,4,1),U=(9,7,7),V=(−6,7,1).
Find the area of the triangle TUV.
area = □
Ergänze die Koordinaten der auf der Parabel markierten Punkte. Die Symmetrieachse der Parabel verläuft parallel zur y-Achse durch den Punkt (2|0).
A - Koordinaten ergänzen
Name GEOMETRY 21: Review for Final Exam Units 1, 2, 3 (First Semester)
Unit 1 - Modeling with Geometry and Definitions (Chapter 1)
Unit 2 - Rigid Motions (Chapter 9)
Per. 1,2,4,7.8
Unit 3 - Geometric Relationships and Properties (Chapters 2, 3, 4, 5, 6 )
True or False
1) Any 2 lines always intersect at one point.
2) Through any 2 points there is exactly one plane.
3) Any 3 points are always coplanar.
4) If AB bisects CD at point E, then AE=EB. Use the diagram at right for questions \#5-9.
5) If \Varangle 2 is a right angle and m4=4x+10 degrees, and m⩽6=8x−4 degrees, find x and m ⩽3. x=m43=
6) If m≮6=y, then write an expression for the m\&BGF.
7) If the m×5=90∘, then name 2 angles that are the complements of ≮4. and
8) If m=5=90∘, name 2 angles that are supplementary, but do not form a linear pair. and
9) HJ⊥FC and AD⊥FC, then AD HJ For #10−12, identify the type of transformation (translation, reflection, rotation).
10)
11)
12) For \#13-16, use the following statement: "Linear pairs are supplementary, adjacent angles."
13) Rewrite the statement as a conditional.
14) Write the converse of the conditional.
15) Write the statement as a biconditional.
16) Is the statement a definition? Explain your reasoning.
```latex
\begin{problem}
a. Find the measure of angle J. Type the answer in the box below. Angle J has a measure of . Explain how you know.
Type your response in the space below. b. Find the measure of angle K. Type the answer in the box below. Angle K has a measure of . Explain how you know.
Type your response in the space below. In parallelogram HIJK, angle H is 45 degrees.
\end{problem}
For 17-18-18, determine the value of x In the given diagram.
17)
18) Mulriple Choice.
19) What type of angles are δ3 and \Varangle .6 ?
(4.) alternate interior
b. alternate exterior
c. same-side interior
d. corresponding
20) If l1∥l2 and m≮1=110∘, then m≮6=
a. 35∘
b. 55∘
c. 70∘
d. 110∘ 21) If l1∥l2 and \mathrm{m} \Varangle 5=75^{\circ}, then \mathrm{m} \Varangle 3=
a. 15∘
b. 75∘
c. 90∘
d. 105∘ 22) If \mathrm{m} \Varangle 5=55^{\circ} and \mathrm{m} \Varangle 4=35^{\circ}, then l1 and l2 -
a. are perpendicular
b. are parallel
c. intersect at an acute angle
d. intersect at an obtuse angle
23) Suppose \Varangle 1 and \Varangle 2 are alternate interior angles formed b parallel lines n and p and transversal t. Which of the following must be true?
a. \Varangle 1 and \Varangle 2 are complementary
b. \Varangle 1 and \Varangle 2 are congruent
c. \Varangle 1 and \Varangle 2 are supplementary
d. \Varangle 1 and \Varangle 2 have a common vertex
24) What is the sum of the measures of the interior angles of a hexagon?
a. 180∘
b. 360∘
c. 540∘
d. 720∘ 25) If the measure of an exterior angle of a regular polygon is 18∘, how many sides does the polygon have?
a. 6
b. 8
c. 15
d. 20
26) The measure of an interior angle of a regular polygon is 140∘. How many sides does it have?
a. 10
b. 9
c. 8
d. 5
27) The measure of an interior angle of a regular polygon is four times the measure of its exterior angle. How many sides does the polygon have?
a. 15
b. 12
c. 10
d. 8
28) If HJ=26, then KL=
a. 13
b. 26
c. 30
d. 52
29) If HJ=3x−1 and KL=x+1, then HJ= .
a. 3
b. 4
c. 8
d. 10
(diagram for \#28-29)
7
NOT TO SCALE To avoid an island, a ship travels 40 kilometres from A to B and then 60 kilometres from B to C.
The bearing of B from A is 080∘ and angle ABC is 115∘.
The duampint ahat a sheteh of a kite SCAly Use a ruler and compasses to construct the kite in the space b The diagonal AB has been drawn for you. Leave in your construction lines.
Determine if the given points form the vertices of a right triangle.
M(2,7),P(4,5), and Q(1,2)
The given points do not form the vertices of a right triangle.
The given points form the vertices of a right triangle.
18. Tiles will be used to cover an area that is 650cm×1,200cm. What is the largest size square tile that can be used so that all the tiles used are full tiles? The largest size square tile that can be used so that all the tiles used are full tiles is
4 Ein 20 m hoher Maibaum steht auf einem Grundstück in der x1x2-Ebene vor einem Hang. Die Ebene E:2x1+6x2+9x3=54 stellt für x1,x2,x3≥0 diesen Hang dar. Im Punkt P(18∣13∣0) wird der 20 m hohe Maibaum senkrecht zum Boden aufgestellt (1LE entspricht 1 m ).
a) Geben Sie die Spurpunkte der Ebene E an und stellen Sie E zusammen mit dem Maibaum im Koordinatensystem dar.
b) Der Maibaum wird auf 19 m Höhe mit einem möglichst kurzen Stahlseil im Hang verankert. Bestimmen Sie die Länge des Stahlseils.
c) Es fällt Sonnenlicht aus der Richtung ⎝⎛−1−4−6⎠⎞ auf den Maibaum. Berechnen Sie die Koordinaten des Punktes S' des Schattens der Maibaumspitze in der Ebene E.
Berechnen Sie außerdem die Koordinaten des Punktes S" des Schattens der Maibaumspitze, wenn der Hang nicht vorhanden wäre. Tragen Sie mithilfe von S' und S′′ den Schatten in der Zeichnung ein. 5 Gegeben sind eine quadratische Pyramide mit den Ecken A(−3∣−3∣0),B(3∣−3∣0),C(3∣3∣0), D(−3∣3∣0) und der Spitze S(0∣0∣9) sowie die Ebene E:3x2+4x3=21.
a) Zeichnen Sie die Pyramide in ein Koordinatensystem.
b) Berechnen Sie die Koordinaten der Schnittpunkte der Pyramidenkanten mit der Ebene E und ergänzen Sie die Schnittfläche in der Zeichnung.
c) Zeigen Sie, dass die Schnittfläche ein Trapez ist. Berechnen Sie den Flächeninhalt des Trapezes.
d) Berechnen Sie den Abstand der Spitze S von der Ebene E.
e) Bestimmen Sie das Volumen der Pyramide und der beiden Teilkörper, in die die Pyramide durch die Ebene E zerlegt wird.
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Eothth grade
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Learning
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Asmalies:
s Which of these relations is a function?
Video (1)
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25
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Submit
Work it out
Not feeling ready yet? These can help:
Identry functions
Lesson; Relations and finctions
Ministerium für Allgemeine und Berufliche Bildung, Wissenschaft, Forschung und Kultur Schleswig-Holstein Schriftliche Abiturprūfung 2023
Name:
Kernfach Mathematik Aufgabe 3: Analytische Geometrie
Die Abbildung zeigt den Körper ABCDEF mit A(6∣3∣0),B(0∣6∣0),C(3∣0∣0),D(6∣3∣6), E(0∣6∣6) und F(3∣0∣12).
a) a1) Untersuchen Sie, ob das Dreieck DEF gleichschenklig ist.
(4 P)
a2) Die Punkte D,E und F liegen in einer Ebene L.
Ermitteln Sie eine Gleichung von L in Koordinatenform.
(4 P)
[zur Kontrolle: L:2x1+4x2+3x3−42=0]⎝⎛001⎠⎞
a3) Bestimmen Sie die Größe des Winkels, den L mit der x1x2-Ebene einschließt.
(3 P)
a4) Berechnen Sie den Abstand des Ursprungs zur Ebene L.
(3P)
a5) Bestimmen Sie eine Gleichung der Schnittgeraden von L mit der x1x2-Ebene in Parameterform.
(3P)
b) b1) Begründen Sie, dass das Viereck ADFC ein Trapez ist.
(2P)
b2) Der Flächeninhalt des Dreiecks ABC kann mit dem Term 6⋅6−21⋅3⋅3−2⋅21⋅3⋅6 berechnet werden. Veranschaulichen Sie diese Tatsache durch geeignete Eintragungen in der Abbildung auf dem Beiblatt.
(3P)
b3) Berechnen Sie das Volumen des Körpers ABCDEF.
(3 P)
2023-M-H3-Analytische Geometrie-L nur für Lehrkräfte
Seite 1 von 5
Q1.
102−62=ADB=8 ADE and AEC are straight lines
DE s paralle to BC.
Angle ABC=90∘AC=10cm.BC=6cm.D is the midpoint of AB.
Work out the area of trapez um BCED.
Determine the direction of the resultant of the following vectors with the x−axis(θx) :
A=3^+7^+8k^B=4^−5^+3k^C=2^+3^−4k^95∘
b. 55.8∘66.3∘43.7∘
Calculate the mass, in tonnes, that is supported by a cylindrical vertical pillar, 1.5 m in diameter, when the stress is 20N/mm2.
A hallow cast-iron vertical cylinder, 3 m long when unloaded, has an outer diameter of 150 mm and an inner diameter of 130 mm . Calculate (a) Maximum load that can be supported at 80 MPa and (b) the decrease in length. Assume 100GPa.
An open-top box is to be constructed from a sheet of tin that measures 42 inches by 24 inches by cutting out squares from each corner as shown and then folding sides. Let V(x) denote the volume of the resulting box.
⟼l=42 inches ⟶
Use the Law of Syllogism to create a new conditional statement from these: If a figure is a square, then it has four congruent sides and four right angles.
Identify the center and radius of these circles: 1. x2+y2=49 2. 5(x2+y2)=125 3. (x+4)2+(y−2)2=9. Find standard forms for: 4. center at origin, radius 53 5. center (17,5), radius 12 6. center (−8,4), contains (−4,2) 7. center (15,7), tangent to x-axis.
Which coordinates for points A′ and B′ show that lines AB and A′B′ are perpendicular? 1. A′:(p,m) and B′:(z,w) 2. A′:(p,m) and B′:(z,−w) 3. A′:(p,−m) and B′:(z,w) 4. A′:(p,−m) and B′:(z,−w)
Determine the transformations to show △QRS≅△TUV. Choose all correct options: A) Reflect across x, then y, then x. B) Reflect across y. C) Rotate 90∘ clockwise, then reflect across y. D) Rotate 180∘, then reflect across x. E) Translate 4 units left.
Figure B is in Quadrant IV. After reflecting, rotating, and translating it, determine if these statements are true or false: a. B′′′ is in Quadrant II.
b. B is in Quadrant III.
c. The x-coordinates of B′′′ are all negative.
d. The y-coordinates of B′′ are all positive. Initial position and shape of B are needed.
Question
Watch Video Triangle EFG is formed by connecting the midpoints of the side of triangle BCD. The lengths of the sides of triangle EFG are shown. What is the length of BD ? Figures not necessarily drawn to scale. Answer Attempt 2 out of 2
BD=□
Submit Answer
Triangle OPQ is formed by connecting the midpoints of the side of triangle LMN. The measures of the interior angles of triangle LMN are shown. Find the measure of ∠MPO. Figures not necessarily drawn to scale.
In the figure below, lines l and k are parallel.
Suppose that m∠2=50∘ and m∠4=30∘. Complete the statements below. We see that ∠1 and ∠4 are
Choose one And since lines l and k are parallel, ∠1 and ∠4 are
Choose one So, m∠1=□∘ 。 We see that ∠2 and ∠5 are
Choose one And since lines l and k are parallel, ∠2 and ∠5 are
Choose one So, m∠5=□∘ 。 By the angle addition property, m∠5+m∠4+m∠3=□
Note that m∠5=50∘ and m∠4=30∘, so m∠3=□ ㅇ.。 Therefore, m∠1+m∠2+m∠3=□]∘. The relationship between ∠1,∠2, and ∠3 is an example of the following rule.
The sum of the interior angle measures of a triangle is □ ○