Question

solve for x. round to the nearest tenth, if necessary. triangle image with right angle at c, angle at d is 72 degrees, side dc is 13, side bc is x

Explanation:

Step1: Find the measure of angle B

In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\). Let \(\angle B=\theta\). Given \(\angle D = 72^{\circ}\), then \(\theta=90^{\circ}-72^{\circ}=18^{\circ}\)

Step2: Use the tangent function

The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\tan B=\frac{CD}{BC}\). We know \(CD = 13\), \(BC=x\), and \(B = 18^{\circ}\)
So, \(\tan(18^{\circ})=\frac{13}{x}\)

Step3: Solve for x

Cross - multiply to get \(x=\frac{13}{\tan(18^{\circ})}\)
Since \(\tan(18^{\circ})\approx0.3249\)
\(x=\frac{13}{0.3249}\approx40.0\)

Answer:

\(x\approx40.0\)