The goal is to find the area of the overall trapezoid (trapezoid MNRS)
To do this, we need three things: A) The bottom horizontal length NR B) The top horizontal length MS C) The height of the trapezoid; the vertical distance from the bottom horizontal piece to the vertical horizontal piece
We don't know (A) but we know (B). The length of MS is 18 ft We don't know (C), but we can find it fairly easily
Area of triangle TPQ = 84 84 = (base)*(height)/2 84 = 12*h/2 84*2 = 12*h 168 = 12*h 12h = 168 h = 168/12 h = 14
The height of triangle TPQ is 14 feet The height of trapezoid MNRS is 14 feet
The hardest part will be figuring out the length of NR
We're told that NP, PQ and QR are in a ratio of 2:1.5:1 this means NP = 2*k PQ = 1.5*k QR = 1*k where k is some positive number
From the diagram we see that PQ is 12 ft, so PQ = 1.5k 1.5k = 12 k = 12/1.5 k = 8 leading to... NP = 2*k = 2*8 = 16 PQ = 1.5*k = 1.5*8 = 12 QR = 1*k = 1*8 = 8
So, NR = NP+PQ+QR NR = 16+12+8 NR = 36
We finally have enough to find the area of the trapezoid area of trapezoid = (height)*(base1+base2)/2 area of trapezoid = (14)*(18+36)/2 area of trapezoid = (14)*(54)/2 area of trapezoid = 756/2 area of trapezoid = 378 square feet
