Geometry: An Interactive Journey to Mastery

/HVVRQ7KH&ODVVL¿FDWLRQRI7ULDQJOHV

Solution

D௘ͽ 7KHWULDQJOHLVFOHDUO\LVRVFHOHV$QGEHFDXVH^2 + 7^2! 72 WKHODUJHVWDQJOHLQWKHWULDQJOHͼ௘RQH

RSSRVLWHDVLGHOHQJWKRI௘ͽLVDFXWH7KXVDOOWKUHHLQWHULRUDQJOHVDUHDFXWHDQGLWLVDQDFXWHLVRVFHOHV

triangle.

E௘ͽ $WULDQJOHIRUH[DPSOHLVLVRVFHOHVDQGREWXVHͼ௘EHFDXVH^2 + 2^2! 32 ௘ͽ$ 2 triangle is

isosceles and right.

F௘ͽ $WULDQJOHGRHVQRWH[LVWͼ௘7KLVLVEHFDXVHLVQRWODUJHU

WKDQ௘ͽ

Example 3

What is the value of yDQGZK\"ͼ௘6HHFigure 15.3௘ͽ

Solution

Angle y is part of a    triangle, and because

22  3522 =, it is a right angle.

Example 4

What are the values of z and w"([SODLQͼ௘6HHFigure 15.4௘ͽ

Solution

7KHWZRWULDQJOHVDUHVLPLODUE\6$6ͼ௘YHUWLFDODQJOHVDQGWKH

VLGHVDQGDQGDQGFRPHLQDUDWLR௘ͽ7KXVz = 36.

Because 15^2 + 36^2 = 39^2 , we have w = 90°.

Study Tip

x ,WLVGLI¿FXOWWRPHPRUL]HWKHIROORZLQJ³,Ia^2 + b^2! c^2 , then the angle is ... .” Instead, hold on to the

image of the three squares on a right triangle and imagine how the area of the largest square changes as

the right angle is decreased to an acute angle or increased to an obtuse angle.

Pitfall

x Save some work and apply the “a^2 + b^2 versus c^2 ́DQDO\VLVWRRQO\RQHDQJOHLQDJLYHQWULDQJOH²

namely, the largest one, opposite the largest side of the triangle.

y^3

2

2

3

Figure 15.3

w

z

24 10 15

(^2639)
Figure 15.4