Question

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?

Answer 1

`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 = (opp)/(hyp)`

`sin theta = 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°`

Answer 2

`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 = (a^2 +b^2 - c^2)/(2ab)`

`cos theta = (5^2+5^2-7^2)/(2xx5xx5)`

`cos theta = 1/50 = 0.02`

Using a calculator or tables you can find the angle:

`theta = 88.85°`

`theta ~~ 89°`