Name:_________________________________________________Date:________________________Hour:___________

Unit 4: Circles Review Key 1. Which of the following doesn’t belong? Circle your choice, and then explain your answer. chord

diameter

tangent

radius

There is no wrong answer here, you just have to be able to defend your answer. Here are some correct answers: The tangent doesn’t belong because it is the only one that is a line. The rest are line segments. The radius doesn’t belong because it is the only one that starts at the center of the circle and goes to the outside. Again, these are just examples. You can do whatever you want, just back it up.

⃡ a tangent line? Explain. 2. Is 𝐴𝐵

If AB is a tangent line, then the angle at B will be a right angle. To tell if it is a right angle or not we can do Pythagorean Theorem and see if it balances. 92 + 152 = 182 81 + 225 = 324 306 = 324 This did not come out equal, so Pythagorean Theorem did not work and the line AB is not tangent to the circle.

3. If point B is a point of tangency, find r.

If AB is a tangent line, then the angle at B is a right angle. Since it is a right angle, Pythagorean Theorem will be balanced, so we can use that to find r. 𝑟 2 + 302 = (𝑟 + 18)2 (r+18)2=(r+18)(r+18) then multiply everything in each

(18+r)

parenthesis by everything in other one. 𝑟 2 + 900 = 𝑟 2 + 18𝑟 + 18𝑟 + 324 We can subtract r2 from both sides. 900 = 18𝑟 + 18𝑟 + 324 Combine like terms. 900 = 36𝑟 + 324 Subtract 324 from both sides. 576 = 36𝑟 Divide both sides by 36. 𝑟 = 16 So the radius is 16.

4. How many different lines could you Sketch a line that would be tangent to the circle below and go through the given point A? two

A

Name:_________________________________________________Date:________________________Hour:___________

5. If B and D are points of tangency, find x. Since BA and DA are tangent lines, the segments are congruent so we can set them equal to each other and then solve. 3𝑥 + 10 = 7𝑥 − 6 Get all the x’s on one side. Minus 3x from both sides. 10 = 4𝑥 − 6 Add 6 to both sides. 16 = 4𝑥 Divide both sides by 4. 4 = 𝑥.

6. You are standing 25 feet from a water tower. You The distance from you to the point of tangency on the tower is 45 feet. What is the radius of the water tower? If that is a tangent line, then the angle is a right angle. Since it is a right angle, Pythagorean Theorem will be balanced, so we can use that to find r. 𝑟 2 + 452 = (𝑟 + 25)2 (r+25)2=(r+25)(r+25) then multiply everything in each parenthesis by everything in other one. 𝑟 2 + 2025 = 𝑟 2 + 25𝑟 + 25𝑟 + 625 We can subtract r2 from both sides. 2025 = 25𝑟 + 25𝑟 + 625 Combine like terms.

r r r+25

2025 = 50𝑟 + 625 Subtract 625 from both sides. 1400 = 50𝑟 Divide both sides by 50. 𝑟 = 28 So the radius is 28.

7. Find x. The angle is an inscribed angle, so the arc is twice the angle measure. Half 44 is 22, so we can say that 14x+2=22 and then solve for x.

44o

14𝑥 + 2 = 22 Minus 2 from both sides.

(14x+2)

14𝑥 = 20 Divide both sides by 14. 𝑥 ≈ 1.4

Name:_________________________________________________Date:________________________Hour:___________ 8. Find x. The angle is a circumscribed angle, so the angle measure is supplementary (adds to 180) with the minor arc. We know that the major arc is 272 o so the minor arc will be 360-272=88o. If the minor arc is 88, then the circumscribed angle will be supplementary

o

272

88o

(adds to 180) with 88, so 180-88=92.

xo

x=92.

9. Find 𝑚 < 𝑆𝑅𝑇 if line 𝑚 is a tangent line.
10. Two students are writing the equation for a circle centered at (4,1) that has a radius of 1. Their answers are shown below. Which student is correct? Explain. Student 1: (𝑥 − 4)2 + (𝑦 − 1)2 = 1 Student : 𝑥 2 + 𝑦 2 − 8𝑥 − 2𝑦 = −16 They are both correct. You can graph both equations in your graphing calculator, and they make the same circle. You could also simplify student 1’s response and see that they are the same. (𝑥 − 4)(𝑥 − 4) + (𝑦 − 1)(𝑦 − 1) = 1 𝑥 2 − 4𝑥 − 4𝑥 + 16 + 𝑦 2 − 𝑦 − 𝑦 + 1 = 1 𝑥 2 + 𝑦 2 − 8𝑥 + 17 + 𝑦 2 − 2𝑦 = 1 𝑥 2 + 𝑦 2 − 8𝑥 − 2𝑦 = −16

11. Write the equation of a circle with a center at (3,-2) that has a radius of 4. The equation for a circle is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 where the center is (h,k) and the radius is the r. (𝑥 − 3)2 + (𝑦 − −2)2 = 42 Simplify. (𝑥 − 3)2 + (𝑦 + 2)2 = 16

Name:_________________________________________________Date:________________________Hour:___________ 12. Write the equation of a circle whose center lies at (2,3) and (6,0) is a point on the circle.

The equation for a circle is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 where the center is (h,k) and the radius is the r. We know the center, but we will need to use Pythagorean Theorem to find the radius because they didn’t give it to us. (𝑥 − 2)2 + (𝑦 − 3)2 = r 2 (see right)

r 4

3 2 + 42 = 𝑟 2 9 + 16 = 𝑟 2 25 = 𝑟 2 5=𝑟

Type equation here.

So now the radius is 5, so the equation is (𝑥 − 2)2 + (𝑦 − 3)2 = 25.

13. Write the equation of a circle whose ends of the diameter fall at (-6,-3) and (2,3). If the diameter falls on those ends then half-way through the line is the radius and center. When you graph it on a grid you can see that the center must be at (-2,0). Then you can use your calculator to make a circle that goes through the center (-2,0) and crosses at one of the diameter end points. Then make you calculator tell you the radius of that circle. It’s 5. So the equation with center (-2,0) and radius 5 will be (𝑥 + 2)2 + 𝑦 2 = 52 Simplify. (𝑥 + 2)2 + 𝑦 2 = 25

14. Write the equation of a circle that is tangent to the x-axis, with the center located at (2,-6).

If it has a center of (2,-6) and is tangent to the x-axis, then it must reach up the x-axis, which would make the radius 6 units long (because it reaches from the center down 6 up to the x-axis). So now we are writing the equation of a circle with a center of (2,-6) and a radius of 6. The equation for a circle is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 where the center is (h,k) and the radius is the r. (𝑥 − 2)2 + (𝑦 − −6)2 = 62 Simplify. (𝑥 − 3)2 + (𝑦 + 6)2 = 36

15. Find the center and the radius of the circle 𝑥 2 + 𝑦 2 + 8𝑥 − 4𝑦 + 4 = 0. Graph it on your calculator and use your calculator to find the center and radius. Open a graph page and then choose “menu”, “graph entry edit”, “relation”. Type in the equation exactly as it’s shown and then hit enter. Choose “menu”, “analyze graph”, “analyze conics” to find the center and the radius. Center: (-4,2) Radius: 4