Chapter 5.5, Problem 37E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# In right triangle X Y Z , X Y = 3 and Y Z = 4 . If V is the midpoint of Y Z ¯ and m ∠ V W Z = 90 ° , find V W .(HINT: Draw X V ¯ . )

To determine

To find:

VW, in the right triangle XYZ, XY=3 and YZ=4. V is the midpoint of YZ and

mVWZ=90°

Explanation

Approach:

Pythagorean Theorem.

In any right triangle, the square of the longer side is equal to the sum of the squares of the other two legs of the triangle.

For the right triangle XYZ with right angle at Z,

XY2=XZ2+YZ2.

Calculation:

Given,

A right triangle XYZ with

XY=3

YZ=4

V is the midpoint of YZ and mâˆ VWZ=90Â°

Since V is the midpoint of YZ

VYÂ¯â‰…VZÂ¯

Thus,

VY=VZ=12Â·YZ

=12Â·4

VY=VZ=2 units.

Draw an auxiliary line segment XVÂ¯ joining X and V.

Apply Pythagorean theorem is the right triangle XYV.

XV2=XY2+YV2

=32+22

=9+4

XV2=13

Apply square root on both sides,

XV=âˆš13 units.

Also, in the right âˆ†XYZ

XZ2=XY2+YV2

=32+42

=9+16

XZ2=25

Apply square root on both sides,

XZ=5 units.

Now, in the right âˆ†VWZ,

VZ2=VW2+WZ2

(2)2=VW2+WZ2 . . . (1)

Again, in the right âˆ†XWV,

XZ2=XW2+VW2

(âˆš13)2=XW2+VW2

13=XW2+VW2 . . . (2)

Since XZ=5 we have

XZ=5

XW+WZ=5

XW+WZ+-WZ=5+(-WZ)

XW+(Â WZ+-WZ=5+(-WZ)

XW=5-W

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