Question

in right $delta xyz$, the length of the hypotenuse $overline{yz}$ is 10 inches and $\tan z=\frac{3}{4}$. what are the lengths, in inches, of the legs $overline{xy}$ and $overline{xz}$? enter your answers in the spaces provided. $xy = $ $xz = $

Explanation:

Step1: Recall tangent definition

$\tan Z=\frac{XY}{XZ}=\frac{3}{4}$, so let $XY = 3x$ and $XZ = 4x$.

Step2: Apply Pythagorean theorem

In right - triangle $XYZ$, by the Pythagorean theorem $XY^{2}+XZ^{2}=YZ^{2}$. Substitute $XY = 3x$, $XZ = 4x$ and $YZ = 10$ into the equation: $(3x)^{2}+(4x)^{2}=10^{2}$.

Step3: Simplify the equation

$9x^{2}+16x^{2}=100$, $25x^{2}=100$, $x^{2}=4$.

Step4: Solve for x

$x = 2$ (we take the positive value since length cannot be negative).

Step5: Find the lengths of the legs

$XY=3x = 3\times2=6$ inches, $XZ = 4x=4\times 2 = 8$ inches.

Answer:

$XY = 6$
$XZ = 8$