Question

An isosceles triangle has a vertical angle of 116 and it's base is 8 cm long calculate it's height

Answer 1

To find the height of the isosceles triangle with a vertical angle of \( 116^\circ \) and a base of \( 8 \) cm, we can use some trigonometry.

1. **Divide the triangle**: Draw a line from the apex (the top point of the triangle) perpendicular to the base. This line bisects the base and creates two right triangles.

2. **Determine the sides**:
- The vertical angle \( A \) of the triangle is \( 116^\circ \).
- Each base angle \( B \) can be calculated as:
\[
B = \frac{180^\circ - 116^\circ}{2} = \frac{64^\circ}{2} = 32^\circ
\]
- The length of the half-base is \( \frac{8 \text{ cm}}{2} = 4 \text{ cm} \).

3. **Use trigonometry**: In one of the right triangles formed, we can use the tangent function to find the height (\( h \)):
\[
\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{4}
\]
\[
\tan(32^\circ) = \frac{h}{4}
\]

4. **Solve for height**:
\[
h = 4 \cdot \tan(32^\circ)
\]

5. **Calculate using a calculator**:
\[
\tan(32^\circ) \approx 0.6249
\]
\[
h \approx 4 \cdot 0.6249 \approx 2.4996 \text{ cm}
\]

Thus, the height of the isosceles triangle is approximately **2.50 cm**.