Question

Find the length of the missing side of the trapezoid below. [asy] size( 100 ) ; real top = 5 ; real bottom = 8 ; real height = sqrt(40) ; real slant = 7 ; pair ll = (-bottom,0) ; pair lr = (0,0) ; pair ur = lr+(0,height) ; pair ul = ur-(top,0) ; draw( ll--lr--ur--ul--cycle ) ; draw(rightanglemark( ur , lr , ll , 20 )) ; draw(rightanglemark( ul , ur , lr , 20 )) ; label( "$"+string(bottom)+"$" , ll--lr , S ) ; label( "$"+string(top)+"$" , ul--ur , N ) ; label( "$"+string(slant)+"$" , ul--ll , W ) ; [/asy] Give your answer in simplified form.

Answer 1

Using the Pythagorean theorem, we confirm that the given slant side length of 7 units is correct for the right-angled trapezoid, so the length of the missing side is also 7 units.

Step-by-step explanation:

We need to find the length of the missing side of a trapezoid. The trapezoid has a top side of 5 units, a bottom side of 8 units, a height of √40 units, and a slant side of 7 units. Since the trapezoid is right-angled at both the bottom and the top, we can use the Pythagorean theorem to find the missing side, which is the other slant side.

To apply the Pythagorean theorem, we consider the slant side as the hypotenuse of a right-angled triangle and the other two sides as the base and the height. The base of this triangle is the difference between the bottom and top lengths of the trapezoid (8 units - 5 units = 3 units).

The Pythagorean theorem is expressed as √(base² + height²) = hypotenuse. Plugging in the known values:

√(3² + (√40)²) = 7

√(9 + 40) = 7

√49 = 7

This equation confirms that the given slant side length of 7 units is correct since the square root of 49 is indeed 7. Therefore, the length of the missing side is also 7 units because it is the same as the given slant side in this right-angled trapezoid.