Question

ABCD is a square with sides of length 10 units. OCD is an isosceles triangle with base CD. OC cuts AB at point Q and OD cuts AB at point P. The area of trapezoid PQCD is 80 sq units. The length of altitude from O of the AOPQ (in units) is

Answer 1

15 units

Explanation:

You want the altitude to base PQ of ∆OPQ given that trapezoid PCDQ is inscribed in square ABCD with PQ part of segment AB, and CP and DQ intersecting at point O. ABCD is a 10-unit square. PCDQ has area 80 square units.

Base

The short base (PQ) of the trapezoid can be found using the area formula:

A = 1/2(b1 +b2)h

80 = 1/2(10 +b2)(10)

160 = (10 +b2)(10)

16 = 10 +b2

b2 = 6 = PQ

Scale factor

Triangle OPQ is similar to triangle OCD, so their dimensions are related by a scale factor. Let x represent the altitude of ∆OPQ. Then (10+x) is the altitude of ∆OCD, and the ratio of these is the same as the ratio of PQ to CD:

x/(10+x) = 6/10

10x = 6(10 +x)

4x = 60 . . . . . . . . subtract 6x

x = 15 . . . . . . . . divide by 4

The altitude of ∆OPQ is 15 units.