Question

in $\triangle xyz$, $angle y=90^\circ$ and $angle x=53^\circ$. $angle zwy=76^\circ$ and $xw=60$. find the length of $zy$ to the nearest integer. figure not necessarily drawn to scale.

Explanation:

Step1: Define variables

Let $ZY = h$ (the side we need to find), and $WY = x$.

Step2: Use trigonometry in $\triangle XYZ$

In right $\triangle XYZ$, $\tan(\angle X) = \frac{ZY}{XY}$. Since $XY = XW + WY = 60 + x$, we get:
$\tan(53^\circ) = \frac{h}{60 + x}$
Rearrange to solve for $x$:
$60 + x = \frac{h}{\tan(53^\circ)}$
$x = \frac{h}{\tan(53^\circ)} - 60$

Step3: Use trigonometry in $\triangle ZWY$

In right $\triangle ZWY$, $\tan(\angle ZWY) = \frac{ZY}{WY}$:
$\tan(76^\circ) = \frac{h}{x}$
Rearrange to solve for $x$:
$x = \frac{h}{\tan(76^\circ)}$

Step4: Set equations for $x$ equal

Equate the two expressions for $x$:
$\frac{h}{\tan(76^\circ)} = \frac{h}{\tan(53^\circ)} - 60$
Use approximate values: $\tan(53^\circ) \approx 1.3270$, $\tan(76^\circ) \approx 4.0108$
Substitute the values:
$\frac{h}{4.0108} = \frac{h}{1.3270} - 60$

Step5: Solve for $h$

Multiply all terms by $4.0108 \times 1.3270 \approx 5.322$ to eliminate denominators:
$1.3270h = 4.0108h - 60 \times 5.322$
$1.3270h = 4.0108h - 319.32$
Rearrange terms to isolate $h$:
$4.0108h - 1.3270h = 319.32$
$2.6838h = 319.32$
$h = \frac{319.32}{2.6838} \approx 119$

Answer:

119