An isosceles triangle has a base of 12 cm with equal sides of 20 cm each. How do you determine the area of this triangle accurate to the nearest square centimeter?

Question

Evelyn Arboleda · Accepted Answer

See a solution process below:

The formula for the area of a triangle is:

#A = 1/2bh#

The base of this isosceles triangle is given as 12cm.

Because the line which bisects and isosceles triangles is at a right angle to the base we can use the Pythagorean Theorem to find the height.

The Pythagorean Theorem states:

#a^2 + b^2 = c^2#

Where:

#a# and #b# are sides of a right triangle.

#c# is the hypotenuse of a right triangle.

In this problem, the hypotenuse, or #c#, is #20"cm"#

One side of the right triangle is the height which we need to solve for.

The other side of the right triangle for this isosceles triangle is #1/2# of the base, or, #6"cm"#

Substituting and solving for #a# gives:

#a^2 + (6"cm")^2 = (20"cm")^2#

#a^2 + 36"cm"^2 = 400"cm"^2#

#a^2 + 36"cm"^2 - color(red)(36"cm"^2) = 400"cm"^2 - color(red)(36"cm"^2)#

#a^2 + 0 = 364"cm"^2#

#a^2 = 364"cm"^2#

#sqrt(a^2) = sqrt(364"cm"^2)#

#a ~= 19"cm"#

Therefore the height is approximately: 19.08cm

We can now substitute into the formula for the area to determine the area of this triangle:

#A = 1/2 xx 12"cm" xx 19"cm"#

#A = 6"cm" xx 19"cm"#

#A = 114"cm"^2# accurate to the nearest square centimeter

Autumn Armstrong · Answer

undefined To determine the area of an isosceles triangle with a base of 12 cm and equal sides of 20 cm each, you can use the following steps:

1. Calculate the height of the triangle using the Pythagorean theorem. Since the triangle is isosceles, the height will be perpendicular to the base and will bisect it.
   
2. Divide the base (12 cm) into two equal parts to find the length of each base segment (6 cm each).

3. Use one of the base segments and one of the equal sides to form a right triangle. The equal side will be the hypotenuse, and the height of the triangle will be one leg of the right triangle.

4. Apply the Pythagorean theorem to find the height. Let's call it $ h $.

5. Once you find the height, use the formula for the area of a triangle, which is $ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $, where the base is 12 cm and the height is the value obtained in the previous step.

6. Calculate the area using the values of the base and height you found.

By following these steps, you can determine the area of the isosceles triangle accurately to the nearest square centimeter.