# High School: Geometry

### Expressing Geometric Properties with Equations HSG-GPE.A.1

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Ah, the circle. The basis of so many delicious baked goods: donuts, pies, cookies, and cakes. Always regular and similar to all of its circle friends. It tries so hard to fit in with the crowd, and yet we insist on labeling it, giving it an equation, shoving it into a 360 degree oven for 12 to 14 minutes, and trying our hardest to let it cool before cramming its deliciousness down our gullets.

Students should know that the equation for a circle is x2 + y2 = r2. You're right, that does look eerily familiar. Sort of like the Pythagorean theorem (a2 + b2 = c2.) Wouldn't it be great if that was all the students needed to know?

If only it were that simple. The equation for a circle is actually (xh)2 + (y – k)2 = r2, where (h, k) is the center of the circle. But when a circle is centered on the origin, or (0, 0), the equation simplifies to x2 + y2 = r2. So actually, comparing it to the Pythagorean theorem comes in quite handy after all.

This circle is centered at (2, 3) with a radius 10, meaning that students have to adjust the left side of the standard circle equation. The adjustment should reflect how we would have to move the circle so that it is centered on the origin; in this case, two to the left and down three. Our equation ends up being (x – 2)2 + (y – 3)2 = r2.

To find the radius, students can draw a right triangle inside the circle and use the Pythagorean theorem to find the length of the hypotenuse. It just so happens that the hypotenuse is also the radius of our circle. Since our hypotenuse here is 10, our equation is (x – 2)2 + (y – 3)2 = 100.

If students are having a hard time remembering whether to do add or subtract their h's and k's, they can simply think, "How would I move the center to the origin?" For example, if their circle was centered at (-5, -17), they'd have to adjust the left side by saying "plus 5" or "plus 17", since they'd have to move the center up and to the right, both of which are positive motions, or as we like to call them, good vibrations.

Perhaps your students find themselves in the lucky situation of having been given an equation and told to find the center and radius of the circle that equation describes. They will need to start by completing the square in the equation so that they can convert it into the standard form.

If they are having a hard time solving these kinds of problems, they might need a quick refresher on how to complete the square. Also, they should note that they're actually completing two squares, since both x and y are squared. For a standard about circles, we're sure using a lot of squares.

#### Drills

1. What is the equation of the circle with center (4, -3) and radius 2?

(x – 4)2 + (y + 3)2 = 4

This equation reflects a center of (4, -3) and a radius of 2. Answer (A) is incorrect because the right side is r rather than r2. The signs in (C) are reversed and would give a circle with center (-4, 3). Answer (D) is also incorrect because the signs are reversed on h and k, and the subtraction sign should be addition.

2. What is the equation of this circle?

(x + 2)2 + (y + 7)2 = ¼

It gives a circle with the center (-2, -7) and radius ½. The x and y variables are reversed in (B), and (C) is incorrect because the x and y variables are negative, not positive, and the radius was not squared. Just the same, the right side of the equation in (D) is incorrect because the radius was not squared either.

3. What is the equation for the circle centered on R with radius RS?

(x – 4)2 + (y – 7)2 = 25

It gives a circle with the center (4, 7) and radius 5. That means our equation should be (x – 4)2 + (y – 7)2 = 25. All the other answer choices are incorrect because either the right side of the equation was not squared or because the positive and negative signs for x and y weren't right.

4. What is the equation for the circle centered on W with radius WZ?

(x – 2)2 + (y – 4)2 = 125

It gives a circle with the center (2, 4) and radius . This gives a circle with the equation (x – 2)2 + (y – 4)2 = 125. All the other answer choices incorrectly use r instead of r2, or place the center of the circle on Z instead of W.

5. Find the equation of a circle centered on point F that has line segment FG as a radius.

(x + 5)2 + (y + 8)2 = 169

Here, we have a circle with a center at (-5, -8) and radius 13. The right side of the equation must be 13 2 = 169, so (A) and (B) are incorrect. Since the center is negative for both x and y variables, the equation should have x + 5 and y + 8 on the left side.

6. What are the center and radius of the circle described by the equation (x + 7)2 + (y – 8)2 = 9?

In the equation, h is -7, k is 8, and the square of the radius is 9. Simply looking at the answer choices for the radius, it's easy to single out (C) as the right answer. All other answers give incorrect radii measurements, and (A) even switches h and k.

7. What are the center and radius of the circle described by the equation x2 + y2 – 18x + 12y + 68 = 0?

Completing the square results in the standard equation (x – 9)2 + (y + 6)2 = 49. The radius is 7, but that doesn't narrow down our choices. Answers (A) and (C) switch the x and y coordinates, and (B) has the signs reversed.

8. What are the center and radius of the circle described by the equation 2x2 + 2y2 + 12x + 20y + 36 = 0?

Completing the square properly will give the equation (x + 3)2 + (y + 5)2 = 16, which features a circle of radius 4 centered on (-3, -5). Answer (A) has is missing the negative signs on h and k, and (C) and (D) forgot to take the square root of 16. Oh, well.

9. Which circle matches the equation 5x2 + 5y2 – 50x + 20y + 100 = 0?

The standard form of this equation is (x – 5)2 + (y + 2)2 = 9, which gives a circle with center (5, -2) and radius 3. The only picture of this circle as it should be is in (A). Errors in the other answer choices include switching the positive and negative signs on the center coordinates and listing the square of the radius.

10. Which circle matches the equation x2 + y2 + x + ½y – ¼ = 0?