Geometry: An Interactive Journey to Mastery

Lesson 19

1. Draw an additional radius and label points as shown in Figure S.19.1.
   +OAC is isosceles, so the two angles labeled a are congruent.
   +OBC is isosceles, so ‘ OBC x a.
   The two vertical angles at M are congruent, and these have
   PHDVXUHVíyíaDQGíxíͼ௘x + a௘ͽ
   &RQVHTXHQWO\íyía íxí xía, from
   which y = 2x follows.


2. y = 20°; x íí ƒz =^12 x = 15°.


3. See Figure S.19.2.
   ^2 + h^2 = 7^2 , so hAB 33, giving 2 33.


4. Draw a common tangent through P as shown in
   Figure S.19.3.
   By the tangent/radius theorem, twice, we see two right angles.
   Thus, ‘  APB 90 90 180°, and A, P, and B are indeed collinear.

y x

a

O a

M

A B

C

Figure S.19.1

4

77

77

4

A

B

h

Figure S.19.2

P

A

B

Figure S.19.3