In ?XYZ, ?X=72° and ?Y=72°. ?XWZ=90° and XY=8.5. Find the length of ZW to the nearest 10th.

Question

In ΔXYZ, ∠X=72° and ∠Y=72°. ∠XWZ=90° and XY=8.5. Find the length of ZW to the nearest 10th.

Answer 1

We draw triangle XYZ with the information given.  I cannot show the triangle here, but I will explain to you how to draw it.  Since angle X and angle Y have the same angle, this triangle is an isosceles triangle.  If we draw a line segment WZ that is perpendicular to XY (because angle XWZ is 90 degrees) , we will have two right triangles that is part of the triangle XYZ.  These triangles are XWZ and YWZ. 

 

You follow?

 

We want to find WZ, so we focus on right triangle YWZ.  Side WY is half the length of XY because WZ is a line bisector of XY.  Therefore, YW is 4.25

 

Now, we have a right triangle with a known angle and a known side.  With this type of triangle, we can use basic trigonometry applications. 

 

The side opposite to the 90 degree angle is the hypotenuse.  The hypotenuse is YZ.  The two legs are WZ and YW.

 

We can use cosine function to find tangent because the YZ is not known.

 

tangent is side opposite of angle divided by side adjacent to angle.

 

tan(72) = WZ / 4.25

 

4.25tan(72) = WZ

 

0.3 = WZ

 

 

Answer 2

Since this is an isoscles triangle, we can automatically say that altitude ZW is also a median. Therefore, both XW and YW have a length of 4.25. Also, we can use trigonometry to solve for ZW.

 

Tan 72 = ZW/4.25

 

ZW = 4.25 * tan 72 = 13.1