# The sides of an isosceles triangle are 5, 5, and 7. How do you find the measure of the vertex angle to the nearest degree?

Dec 7, 2016

89° to the nearest degree.

#### Explanation:

The base of the triangle 7 can be divided in half by a line of symmetry of the isosceles triangle, which will bisect the vertex angle. This creates two right triangles:

Each with a base of 3.5 and a hypotenuse of 5.

The side opposite the half of the vertex angle is 3.5, the hypotenuse is 5.

The sine function can be used to find the angle.

$\sin \theta = \frac{o p p}{h y p}$

$\sin \theta = \frac{3.5}{5} = 0.7$

Use the inverse sin function or a table of trig functions to find the corresponding angle . (Arcsin)

arcsin 0.7 = 44.4°

Remember that this is the value of half of the vertex angle so double the value to find the vertex angle.

2 xx 44.4 = 88.8 °

rounded off to the nearest whole degree = 89°

Dec 9, 2016

theta ~~ 89°

#### Explanation:

As all 3 sides of the triangle are known, the cosine rule can be used to find the vertex angle directly.

$\cos \theta = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{2 a b}$

$\cos \theta = \frac{{5}^{2} + {5}^{2} - {7}^{2}}{2 \times 5 \times 5}$

$\cos \theta = \frac{1}{50} = 0.02$

Using a calculator or tables you can find the angle:

theta = 88.85°

theta ~~ 89°