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Math  /  Geometry

Question13) Ends of a diameter: (−17,−9)(-17,-9) and (−19,−9)(-19,-9) 14) Ends of a diameter: (−3,11)(-3,11) and (3,−13)(3,-13) 15) Center: (−15,37)(-15,3 \sqrt{7}) 16) Center: (−11,−14)(-11,-14)
Area: 2π2 \pi Area: 16π16 \pi 17) Center: (−5,12)(-5,12) 18) Center: (15,14)(15,14)
Circumference: 8π8 \pi Circumference: 2π152 \pi \sqrt{15} 19) Center: (2,−5)(2,-5)
Point on Circle: (−7,−1)(-7,-1) 20) Center: (14,17)(14,17)
Point on Circle: (15,17)(15,17) 21) Center: (−15,9)(-15,9) 22) Center: (−2,12)(-2,12)
Tangent to x=−17x=-17 Tangent to x=−5x=-5

Studdy Solution

STEP 1

1. The problems involve finding the radius or diameter of a circle given certain information.
2. The formulas for the area and circumference of a circle are known.
3. The distance formula is used to find the length of a diameter.
4. The relationship between the radius and the tangent line is known.

STEP 2

1. Calculate the radius from the ends of a diameter.
2. Calculate the radius from the area.
3. Calculate the radius from the circumference.
4. Calculate the radius from a point on the circle.
5. Determine the radius from a tangent line.

STEP 3

For problems 13 and 14, calculate the radius from the ends of a diameter using the distance formula.
For problem 13: - Points: (−17,−9)(-17,-9) and (−19,−9)(-19,-9) - Use the distance formula: d=(−19−(−17))2+(−9−(−9))2 d = \sqrt{(-19 - (-17))^2 + (-9 - (-9))^2} d=(−2)2+02=4=2 d = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2 - The diameter is 22, so the radius r=22=1r = \frac{2}{2} = 1.
For problem 14: - Points: (−3,11)(-3,11) and (3,−13) (3,-13) - Use the distance formula: d=(3−(−3))2+(−13−11)2 d = \sqrt{(3 - (-3))^2 + (-13 - 11)^2} d=62+(−24)2=36+576=612=617 d = \sqrt{6^2 + (-24)^2} = \sqrt{36 + 576} = \sqrt{612} = 6\sqrt{17} - The diameter is 6176\sqrt{17}, so the radius r=6172=317r = \frac{6\sqrt{17}}{2} = 3\sqrt{17}.

STEP 4

For problems 15 and 16, calculate the radius from the area.
For problem 15: - Area: 2π2\pi - Use the area formula: πr2=2π \pi r^2 = 2\pi r2=2 r^2 = 2 r=2 r = \sqrt{2}
For problem 16: - Area: 16π16\pi - Use the area formula: πr2=16π \pi r^2 = 16\pi r2=16 r^2 = 16 r=4 r = 4

STEP 5

For problems 17 and 18, calculate the radius from the circumference.
For problem 17: - Circumference: 8π8\pi - Use the circumference formula: 2πr=8π 2\pi r = 8\pi r=4 r = 4
For problem 18: - Circumference: 2π152\pi\sqrt{15} - Use the circumference formula: 2πr=2π15 2\pi r = 2\pi\sqrt{15} r=15 r = \sqrt{15}

STEP 6

For problems 19 and 20, calculate the radius from a point on the circle.
For problem 19: - Center: (2,−5)(2,-5), Point: (−7,−1)(-7,-1) - Use the distance formula: r=(−7−2)2+(−1−(−5))2 r = \sqrt{(-7 - 2)^2 + (-1 - (-5))^2} r=(−9)2+42=81+16=97 r = \sqrt{(-9)^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97}
For problem 20: - Center: (14,17)(14,17), Point: (15,17)(15,17) - Use the distance formula: r=(15−14)2+(17−17)2 r = \sqrt{(15 - 14)^2 + (17 - 17)^2} r=12+02=1=1 r = \sqrt{1^2 + 0^2} = \sqrt{1} = 1

STEP 7

For problems 21 and 22, determine the radius from a tangent line.
For problem 21: - Center: (−15,9)(-15,9), Tangent to x=−17x = -17 - The radius is the horizontal distance from the center to the tangent line: r=∣−15−(−17)∣=∣2∣=2 r = |-15 - (-17)| = |2| = 2
For problem 22: - Center: (−2,12)(-2,12), Tangent to x=−5x = -5 - The radius is the horizontal distance from the center to the tangent line: r=∣−2−(−5)∣=∣3∣=3 r = |-2 - (-5)| = |3| = 3

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