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?

Question

Charles Autterson · Accepted Answer

#89°# to the nearest degree.

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°#

Dylan Arnold · Answer

#theta ~~ 89°#

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°#

Quinn Anderson · Answer

undefined To find the measure of the vertex angle in an isosceles triangle, you can use the formula for finding the interior angles of a triangle. Since it's an isosceles triangle, two sides are equal. You can use the Law of Cosines to find the measure of the vertex angle. The formula is:

$c^2 = a^2 + b^2 - 2ab\cos(C)$

Substitute the values where $a = b = 5$ and $c = 7$, then solve for $\cos(C)$. After finding $\cos(C)$, use the inverse cosine function to find the measure of angle $C$. Once you find the measure of angle $C$, which is the vertex angle, round it to the nearest degree.