Question

the length of the base of an isosceles triangle is 35.34 inches. each base angle is 25.66°. find the length of each of the two equal sides of the triangle. (hint: divide the triangle into two right triangles.) the other two sides of the triangle have lengths of \\(\square\\) inches. (simplify your answer. type an integer or decimal rounded to the nearest hundredth as needed.)

Explanation:

Step1: Divide the isosceles triangle

When we divide the isosceles triangle into two right triangles by drawing an altitude from the apex to the base, the altitude bisects the base. So the length of the adjacent side (let's call it \(x\)) of the right triangle (with respect to the base angle) is half of the base of the isosceles triangle.
The base of the isosceles triangle is \(b = 35.34\) inches, so \(x=\frac{35.34}{2}= 17.67\) inches.

Step2: Use trigonometric ratio (cosine)

We know that in a right triangle, \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 25.66^{\circ}\), the adjacent side is \(x = 17.67\) inches, and the hypotenuse is the length of the equal side of the isosceles triangle (let's call it \(l\)).
So \(\cos(25.66^{\circ})=\frac{17.67}{l}\)
We can solve for \(l\) by rearranging the formula: \(l=\frac{17.67}{\cos(25.66^{\circ})}\)

Step3: Calculate the value

First, find the value of \(\cos(25.66^{\circ})\). Using a calculator, \(\cos(25.66^{\circ})\approx0.902\) (rounded to three decimal places).
Then \(l=\frac{17.67}{0.902}\approx19.59\) (rounded to the nearest hundredth)

Answer:

\(19.59\)