GMAT Geometry Chapter 4: Circles

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GMAT Geometry Guide

		Chapter 1: Angles and Lines

		Chapter 2: Intersecting Angles

		Chapter 3 Triangles

		Chapter 4: Circles

		Chapter 5: Perimeters & Areas

		Chapter 6: Solids

		Chapter 7: Coordinate Geometry

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The diameter (d) of a circle is twice the radius (r). A circle's circumference is d or 2r ( = 3.14 or 22/7- which is approximately 3.14).

A central angle has its vertex at the center of a circle, and its measure equals the measure of the arc it intercepts (in degrees). For example, if AOB = 60, then the measure of arc AB is 60°, or 60/360 = 1/6 of the circle's circumference.

Circumference = 2

r =

AOB = arc AB

An inscribed angle has its vertex on the circle itself, and its measure is 1/2 of the measure of the arc it intercepts:

ACB = 1/2 arc AB.

A line that just touches a circle is called a tangent. It is perpendicular to the radius drawn to the point of touching.

ABC is a right triangle if CB is the diameter. A triangle inscribed in a circle is a right triangle if one of its sides is a diameter. Obviously,

A has its vertex on the circle, and it intercepts half of the circle so that

A = 180°/ 2 = 90°.

Note: if an inscribed triangle has a leg as the diameter, that leg is the hypotenuse.

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Example 1 (medium)

What arc length is intercepted by an inscribed angle of 42° on a circle with r = 12 (where = 3.14 = 22/7)?

Solution

The 42° inscribed angle intercepts 1/2(arc°) or arc° = 84°; that is, 84/360 of the circle is intercepted by the angle. The circumference is 2r = 24 so that the arc length is, using = 22/7,

arc length = 84/360 × 24 =
factor out the 12s in 84 and 360, factor 24 into 6 and 4, and convert into 22/7.

(7 × 12)/(30 × 12) × (6 × 4) 22/7 = 88/5 = 17.6

Example 2 (easy)

A triangle is inscribed in a circle with shorter sides 6 and 8 units long. If the longer side is a diameter, find the length of the diameter.

Solution

A triangle so inscribed (with one side a diameter) is a right triangle. Consequently,

d = 6 + 8 = 36 + 64 = 100; therefore d = 10. Or, you could have just seen that 6, 8 is double 3, 4.

Example 3 (hard)

A certain clock has a minute hand that is exactly 3 times as long as it's hour hand. Point C is at the tip of the minute hand, and point D is at the tip of the hour hand. What is the ratio of the distance that point C travels to the distance that point D travels in 6 hours?

A. 3:1
B. 6:1
C. 12:1
D. 18:1
E. 36:1

Solution

In 6 hours, the point C on the minute hand travels 6 circumferences (where point C to the middle of the clock is the radius of its circle). The point D on the hour hand only travels half way round the clock, half a circumference (where point D to the middle of the clock is the radius of it's circle).

Since the minute hand is 3 times as long as the hour hand, let the distance between point C and the center of the clock be 3r and the distance from point D to the center be r.

Point C travels 6 × 2(3r) = 36r
Point D travels 0.5 × 2r = r

36r
r

Thus, the ratio of the distance that point C travels to the distance that point D travels in 6 hours is 36:1. The correct answer is E.

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