SAT Mc Graw Hill 2011

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 rfrom the center.


• The radiusis the distance from the center
  to any point on the circle.
  Radiusmeans rayin Latin; a radius comes from
  the center of the circle like a ray of light from the
  sun.


• The diameteris twice the radius: d= 2 r.
  Dia- means throughin Latin, so the diameter is
  a segment that goes all the way through thecircle.

The Circumference and Area

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

• Remember that the formulas for circumfer-
  ence and area are given in the reference in-
  formation 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:” area=πr^2.

Tangents

Lesson 8: Circles

diameterradius

tangent

radius

A tangent is a line that touches (or intersects) the cir-

cle 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.

M

P

R

l

M

P

R

l

7

5

400 McGRAW-HILL’S SAT

Example:

In the diagram above, point Mis 7 units away

from the center of circle P.If line lis tangent to the

circle and MR=5, what is the area of the circle?

First, connect the dots. Draw MPand PRto make a

triangle.

Since PRis a radius and MRis a tangent, they are

perpendicular.

Since you know two sides of a right triangle, you

can use the Pythagorean theorem to find the third

side: 5^2 +(PR)^2 = 72

Simplify: 25 +(PR)^2 = 49

Subtract 25: (PR)^2 = 24

(PR)^2 is the radius squared. Since the area of the

circle is πr^2 , it is 24π.