Question

solve for the value of
a)
(there is a right triangle w v u, with a right angle at v. angle at w is 68 degrees. the length of v u is 96, and the length of the segment from w to the point on w u (marked as x) is to be found. there are two lines crossing inside the triangle, indicating some mid - segment or angle - bisector - like feature, but the main elements are the right angle at v, angle 68° at w, side v u = 96, and we need to solve for x on side w u.)

Explanation:

Step1: Identify triangle type & angles

The figure is right triangle $\triangle WVU$, $\angle V=90^\circ$, $\angle W=68^\circ$, so $\angle U = 180^\circ - 90^\circ - 68^\circ = 22^\circ$. The tick marks mean $WU$ (side $x$) and $VU$ are not equal; instead, we use trigonometric ratios for right triangles. We know $\cos(\angle W) = \frac{WV}{WU}$, but we can use $\sin(\angle U) = \frac{WV}{x}$, or more directly: $\cos(68^\circ) = \frac{WV}{x}$ and $\sin(68^\circ) = \frac{VU}{x}$. Since $VU=96$, use sine:
$\sin(68^\circ) = \frac{96}{x}$

Step2: Rearrange to solve for $x$

Isolate $x$ by cross-multiplying:
$x = \frac{96}{\sin(68^\circ)}$

Step3: Calculate the value

Use $\sin(68^\circ) \approx 0.9272$:
$x \approx \frac{96}{0.9272} \approx 103.54$

Answer:

$x \approx 104$ (rounded to nearest whole number) or $\boldsymbol{\frac{96}{\sin(68^\circ)} \approx 103.5}$