## SAT Test Prep

## CHAPTER 10

ESSENTIAL GEOMETRY SKILLS

### Lesson 8: Circles

**Circle Basics**

Okay, we all know a circle when we see one, but it often helps to know the mathematical definition of a circle.

• A circle is all of the points in a plane that are a certain distance *r* from the center.

• The *radius* is the distance from the center to any point on the circle.

*Radius* means *ray* in Latin; a radius comes from the center of the circle like *a ray of light from the sun*.

• The *diameter* is twice the radius:

*Dia*- means *through* in Latin, so the diameter is a segment that goes *all the way through the* circle.

**The Circumference and Area**

It”s easy to confuse the circumference formula with the **area** formula, because both formulas contain the same symbols arranged differently: *circumference* = 2*πr* and *area* = *πr*^{2}. There are two simple ways to avoid that mistake:

• Remember that the formulas for circumference and area are given in the reference information at the beginning of every math section.

• Remember that area is always measured in square units, so the area formula is the one with the “square: ” .

**Tangents**

A *tangent is a line that touches (or intersects) the circle at only one point*. Think of a plate balancing on its side on a table: the table is like a tangent line to the plate.

A tangent line is always perpendicular to the radius drawn to the point of tangency.

Just think of a bicycle tire (the circle) on the road (the tangent): notice that the center of the wheel must be “directly above” where the tire touches the road, so the radius and tangent must be perpendicular.

**Example:**

In the diagram above, point *M* is 7 units away from the center of circle *P*. If line *l* is tangent to the circle and , what is the area of the circle?

First, connect the dots. Draw *MP* and *PR* to make a triangle.

Since *PR* is a radius and *MR* is a tangent, they are perpendicular.

Since you know two sides of a right triangle, you can use the Pythagorean theorem to find the third

(*PR*)^{2} is the radius squared. Since the area of the circle is π*r*^{2}, it is 24π.

**Concept Review 8: Circles**

__1.__ What is the formula for the circumference of a circle?

__2.__ What is the formula for the area of a circle?

__3.__ What is a tangent line?

__4.__ What is the relationship between a tangent to a circle and the radius to the point of tangency?

__5.__ In the figure above, is a tangent to circle *C*, , and . What is the circumference of circle *C*?

__6.__ In the figure above, *P* and *N* are the centers of the circles and are 6 centimeters apart. What is the area of the shaded region?

**SAT Practice 8: Circles**

**1**__.__ Two circles, *A* and *B*, lie in the same plane. If the center of circle *B* lies on circle *A*, then in how many points could circle *A* and circle *B* intersect?

I. 0

II. 1

III. 2

(A) I only

(B) III only

(C) I and III only

(D) I and III only

(E) I, II, and III

**2**__.__ What is the area, in square centimeters, of a circle with a circumference of 16π centimeters?

(A) 8π

(B) 16π

(C) 32π

(D) 64π

(E) 256π

**3**__.__ Point *B* lies 10 units from point *A*, which is the center of the circle of radius 6. If a tangent line is drawn from *B* to the circle, what is the distance from *B* to the point of tangency?

__Note__: Figure not drawn to scale.

**4**__.__ In the figure above, and are tangents to circle *C*. What is the value of *m*?

**5**__.__ In the figure above, circle *A* intersects circle *B* in exactly one point, is tangent to both circles, circle *A* has a radius of 2, and circle *B* has a radius of 8. What is the length of ?

**Answer Key 8: Circles**

**Concept Review 8**

__1.__ circumference = 2π *r*

__3.__ A tangent line is a line that intersects a circle at only one point.

__4.__ Any tangent to a circle is perpendicular to the radius drawn to the point of tangency.

__5.__ Draw to make a right triangle, and call the length of the radius *r*. Then you can use the Pythagorean theorem to find *r:*

The circumference is 2π*r*, which is .

__6.__ Draw the segments shown here. Since *PN* is a radius of both circles, the radii of both circles have the same length. Notice that *PN, PR, RN, PT*, and *NT* are all radii, so they are all the same length; thus, Δ *PNT* and Δ*PRN* are equilateral triangles and their angles are all 60° . Now you can find the area of the left half of the shaded region. This is the area of the sector minus the area of Δ*RNT*. Since ∠*RNT* is 120°, the sector is 120/360, or , of the circle. The circle has area 36°, so the sector has area 12π.Δ*RNT* consists of two 30°-60°-90° triangles, with sides as marked, so its area is . Therefore, half of the original shaded region is , and the whole is .

**SAT Practice 8**

__1.__ **E** The figure above demonstrates all three possibilities.

__2.__ **D** The circumference . Dividing by 2π. gives . Area .

__3.__ **8** Draw a figure as shown, including the tangent segment and the radius extended to the point of tangency. You can find *x* with the Pythagorean theorem:

__4.__ **45** Since and are tangents to the circle, they are perpendicular to their respective radii, as shown. The sum of the angles in a quadrilateral is 360°, so *m* = 3*m* = 90 + 90 = 360 Simplify: 4*m* = 180 360 Subtract 180: 4*m* = 180 Divide by 4:*m* 45

__5.__ **8** Draw the segments shown. Choose point *E* to make rectangle *ACDE* and right triangle *AEB*. Notice that because opposite sides of a rectangle are equal. You can find *AE* with the Pythagorean theorem: