The quadrilateral abcd has area of 58 in2 and diagonal ac = 14.5 in. find the length of diagonal bd if ac ⊥ bd.

The correct answer is:

8 in.

Explanation:

The quadrilaterals with perpendicular diagonals are a square and a rhombus.  In both of these, the diagonals are perpendicular bisectors of one another.

Since a square is a rhombus, we will use the formula for the area of a rhombus to solve this problem:

A = (diagonal 1 × diagonal 2)/2

Let diagonal 1 = AC and diagonal 2 = BD.  We know that AC = 14.5 and the area, A, is 58:
58 = (14.5 × BD)/2

Multiply both sides by 2:
58×2 = ((14.5×BD)/2)×2
116 = 14.5×BD

Divide both sides by 14.5:
116/14.5 = 14.5×BD/14.5
8 = BD

The measure of the second diagonal, BD, is 8 inches.