Question

1. 3 In right AXYZ, the length of the hypotenuse YZ is 85 inches and tan Z= 3/4 What is the length, in inches, of the leg XY?

Answer 1

We have a right triangle XYZ.

The length of the hypotenuse is YZ=85.

We also know that the tangent of Z is 4.

We have to find the length of XY.

We can start by drawing the triangle and writing the data:

The tangent of an angle can be related with the sides by the following trigonometric ratio:

`\tan (Z)=\frac{\text{Opposite}}{\text{Adyacent}}=(XY)/(XZ)=(3)/(4)`

We can not find the value of the legs from the trigonometric ratio, but we have a proportion between them. We can write the previous result as:

`\begin{gathered} (XY)/(XZ)=(3)/(4) \\ XZ=(4)/(3)\cdot XY \end{gathered}`

Now we can relate XY with the hypotenuse YZ using the Pythagorean theorem:

`\begin{gathered} XY^2+XZ^2=YZ^2 \\ XY^2+((4)/(3)XY)^2=YZ^2 \\ XY^2+(16)/(9)XY^2=YZ^2 \\ ((16)/(9)+1)XY^2=YZ^2 \\ (16+9)/(9)XY^2=YZ^2 \\ (25)/(9)XY^2=YZ^2 \\ XY^2=(9)/(25)YZ^2 \\ XY=\sqrt[]{(9)/(25)YZ^2} \\ XY=(3)/(5)YZ \\ XY=(3)/(5)\cdot85 \\ XY=51 \end{gathered}`

Answer: the length of the leg XY is 51 inches.