An isosceles triangle has a base of 20 cm. The vertex angle is 82 . Find the length of
Each of the two equal sides of the triangle.
16.24
To find the length of each of the two equal sides of the isosceles triangle, we need to use the properties of an isosceles triangle.
In an isosceles triangle, the two equal sides are opposite to the two equal angles.
Given:
Base = 20 cm
Vertex angle = 82°
Step 1: To find the equal angles, we subtract the vertex angle from 180° (since the sum of the angles in a triangle is 180°).
180° - 82° = 98°
Step 2: Since the triangle is isosceles, the other two angles are equal. So, each of the equal angles is equal to half the difference of the sum of the angles from 180°.
(180° - 98°) / 2 = 41°
Step 3: Now, we can find the length of each of the two equal sides using the sine function.
In a triangle, sin(angle) = opposite / hypotenuse.
In this case, the opposite side is half of the base (since the base is split into two equal sides).
sin(41°) = (1/2)(20 cm) / length of equal side
Simplifying, we have:
sin(41°) = 10 cm / length of equal side
Step 4: We can solve for the length of each equal side by rearranging the equation:
length of equal side = 10 cm / sin(41°)
Using a calculator to find the sine of 41° and dividing 10 cm by the sine value, we get:
length of equal side ≈ 15.518 cm
Therefore, the length of each of the two equal sides of the isosceles triangle is approximately 15.518 cm.
To find the length of each of the two equal sides of an isosceles triangle, we can use the properties of the triangle.
In an isosceles triangle, the two equal sides are opposite the two equal angles. Since one of the angles is given (the vertex angle), we can find the measure of the other equal angle by using the fact that the sum of the angles in a triangle is 180 degrees.
To find the measure of the other angle, we subtract the vertex angle from 180 degrees:
180 degrees - 82 degrees = 98 degrees
Now we have two equal angles in the isosceles triangle, which means the two equal sides are opposite these angles. Let's call the length of each equal side "x".
We can use the sine function to relate the side length "x" to the angle measure. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse (in this case the side opposite the angle is "x" and the hypotenuse is the base of the triangle, which is 20 cm).
Using the sine function:
sin(98 degrees) = x / 20 cm
To find the value of "x", we rearrange the equation:
x = sin(98 degrees) * 20 cm
Now, we need to evaluate sin(98 degrees). However, most calculators and trigonometric tables provide values for the sine function based on angles in degrees. So, we can use a scientific calculator or an online calculator to find the sine of 98 degrees, which is approximately 0.9063.
Plugging in the values, we get:
x = 0.9063 * 20 cm
x ≈ 18.13 cm
Therefore, each of the two equal sides of the isosceles triangle is approximately 18.13 cm long.
if each side is s, then
10/s = sin41