A trapezoid has parallel sides 13 and 21 cm long. The longer of the two nonparallel sides is 17 cm and the shorter is perpendicular to a parallel side. What is the area of the trapezoid?
make a sketch.
split the figure into a rectangle and a right-angled triangle.
in the triangle the base will be 21-13 or 8 cm
and the hypotenuse is 17
let the height be h
h^2 + 8^2 = 17^2
h^2 = 225
h = 15
area = rectangle + triangle
= 13(15) cm^2 + (1/2)(8)(15)
= 255 cm^2
To find the area of a trapezoid, we can use the formula:
Area = (1/2) × (sum of the lengths of the parallel sides) × (distance between the parallel sides)
In this case, the lengths of the parallel sides are 13 cm and 21 cm. The longer nonparallel side is 17 cm and the shorter side is perpendicular to a parallel side. Let's label the shorter side as x and the distance between the parallel sides as h.
Since the shorter side is perpendicular to a parallel side, it forms a right triangle with the longer nonparallel side. We can use the Pythagorean theorem to find the value of x.
Pythagorean Theorem: a^2 + b^2 = c^2
In this case, a = x, b = h, and c = 17 cm (the longer nonparallel side).
x^2 + h^2 = 17^2
x^2 + h^2 = 289
Since the problem doesn't provide us with the value of h, we can't solve the equation directly. However, we can make use of another property of a trapezoid.
If we draw a line segment from the end of the shorter side to the parallel side with a length of h, it creates a right triangle. Let's label one of the legs of this triangle as h (same as before), and the other leg as y.
Now, we have two similar triangles (same shape but different sizes) – the large trapezoid and the small right triangle. Both triangles share an angle, which means the ratio of their corresponding side lengths is the same.
Using this similarity, we can set up a proportion:
x/h = 21/h + y
x = 21 + (h * y)/h
x = 21 + y
Now we have a system of two equations:
x^2 + h^2 = 289
x = 21 + y
Solving these two equations simultaneously will give us the values of x and h.
Now that we have the values of x and h, we can calculate the area of the trapezoid using the formula mentioned earlier.
Area = (1/2) × (13 + 21) × h
Therefore, the area of the trapezoid can be found by following these steps to get the values of x and h, and then using the area formula.