the base of an isosceles triangle is 20cm and its perimeter is 42cm. find its area
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To find the area of an isosceles triangle, we need to know the lengths of the two equal sides, as well as the base. However, in the given question, only the base and perimeter are mentioned.
To proceed with finding the area, we first need to find the lengths of the two equal sides.
Perimeter = sum of all three sides of the triangle
42cm = base + side1 + side2
In an isosceles triangle, the base is the different side, and the other two sides are equal. So, for this triangle, the equation becomes:
42cm = 20cm + side1 + side1
42cm = 20cm + 2 * side1
2 * side1 = 42cm - 20cm
2 * side1 = 22cm
To find the length of each equal side, divide both sides of the equation by 2:
side1 = 22cm / 2
side1 = 11cm
Now that we have determined the lengths of the two equal sides (11cm each) and the base (20cm), we can find the area of the triangle.
The formula to calculate the area of any triangle is:
Area = (1/2) * base * height
In an isosceles triangle, the height is the perpendicular distance from the base to the top vertex. To calculate the height, we can use the Pythagorean theorem.
Since the triangle is isosceles, it will have a line of symmetry. Hence, if we draw a line perpendicular to the base from the top vertex, it will intersect the base at its midpoint. So, the two equal sides of the triangle are divided into two equal segments of length x, where x is the height.
Using the Pythagorean theorem:
(11cm / 2)^2 + x^2 = 11cm^2
Solve for x:
(121cm^2 / 4) + x^2 = 121cm^2
121cm^2 / 4 + x^2 = 121cm^2
x^2 = 121cm^2 - 121cm^2 / 4
x^2 = (484cm^2 - 121cm^2) / 4
x^2 = 363cm^2 / 4
x^2 = 90.75cm^2
x ≈ √90.75cm^2
x ≈ 9.54cm
Now we have the height (x = 9.54cm) and the base (20cm) of the triangle. We can find the area using the formula:
Area = (1/2) * base * height
Area = (1/2) * 20cm * 9.54cm
Area ≈ 191.05cm^2
Therefore, the approximate area of the isosceles triangle is 191.05 square centimeters.