I need help....?

Question

How do you find the height of a trapezoid that doesn't show the number of its height but sides and a 45 degree?

Answer 1

Hi,

Suppose you had an isosceles trapezoid with the 2 bases having lengths of 18" and 24". There is also a 45° angle on each side.
......18' base
.._________
/__________\45° angle
...24" base

If we have this, find the difference in the length of the bases and divide that number in half. For this problem, 24 - 18 = 6 and half of 6 is 3. That means the longer base sticks out 3" more on both the left and the right sides. That little "extra part is actually the base of a 45-45-90° right triangle with a base of 3". It so happens that the height of the triangle is also 3" because in a 45-45-90° triangle, the 2 legs are the same length. That means that the triangle's height of 3" is also the height of the trapezoid. So to find its area we would use the formula A = ½h(b1 + b2). this becomes

A = ½(3)(18 + 24)
A = ½(3)(42) = 63 sq. in.

Suppose instead you had a trapezoid with a 45° on one end, but the other side was perpendicular to the bases. If the bases were 18" and 24", then your trapezoid would be:

18 inch base
_________
|_________\45° angle
24 inch base

In this case the longer base is 6" longer than the shorter base, all on one end. This 6" is the base of a 45-45-90° right triangle. It so happens that the height of the triangle is also 6" because in a 45-45-90° triangle, the 2 legs are the same length. That means that the triangle's height of 6" is also the height of the trapezoid. So to find its area we would use the formula A = ½h(b1 + b2). this becomes

A = ½(6)(18 + 24)
A = ½(6)(42) = 126 sq. in.

If you knew the 2 bases were 18" and 24" in length, and the length of the non-parallel side beside the 45° angle is 10", you can find the height of the trapezoid. The height of the trapezoid will be half the length of the slanted side times √2.
So with a slanted side of 10", the height would be half of 10, or 5√2. So to find its area we would use the formula A = ½h(b1 + b2). this becomes

A = ½(5√2)(18 + 24)
A = ½(5√2)(42) = 105√2 sq. in.



I hope those examples help!! :-)

Answer 2

In a 45-45-90 right triangle, the hypotenuse is √2 times the base or the height, so divide the side you've got by √2.