The lengths of the three sides of an isosceles triangle are 24", 24", and 14". Find the measure of each length of each angle. Then find the area of the triangle.
to get the vertex angle θ,
sin(θ/2) = 7/24
Once you have θ, the other two angles are equal, and they all three add up to 180.
To find the measures of the angles in an isosceles triangle, you can use the fact that the base angles (the angles opposite the equal sides) are congruent.
Step 1: Determine the measures of the base angles.
Since the triangle is isosceles, the base angles are congruent. Let's denote the measure of each base angle as "x".
Therefore, we have two equations:
x + x + 14° = 180° (The sum of the angles in a triangle is 180°)
2x = 166°
x = 83°
Step 2: Determine the measure of the remaining angle.
The remaining angle, which is the apex angle (the angle opposite the base), can be found by subtracting the sum of the base angles from 180°.
Apex angle = 180° - (83° + 83°) = 180° - 166° = 14°
So, the measure of each base angle is 83°, and the measure of the apex angle is 14°.
To find the area of the triangle, we can use the formula for the area of a triangle when the lengths of the three sides are known. The formula is known as Heron's formula:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
Where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter (half of the perimeter):
s = (a + b + c) / 2
In our case, a = b = 24" and c = 14".
So, the semi-perimeter s = (24 + 24 + 14) / 2 = 31".
Now, we can calculate the area:
Area = sqrt(31 * (31 - 24) * (31 - 24) * (31 - 14))
= sqrt(31 * 7 * 7 * 17)
≈ sqrt(20623)
≈ 143.50 square inches (rounded to two decimal places)
Therefore, the area of the isosceles triangle is approximately 143.50 square inches.