Depth · one concept from first principles · step 2 of 4

Trapezoid minus two right triangles, in letters

For the points $P(a, b)$ and $Q(-2a, 3b)$ , where $a>0$ and $b>0$ , the area of triangle $OPQ$ is $15$ . Find the value of the product $ab$ of the constants $a$ and $b$ . ( $O$ is the origin.)

GeometryExpressions

Your answer

How to solve

Strategy Identify Subproblems — No side of triangle OPQ lies along a grid line, so split the job into pieces whose areas are easy: drop perpendiculars from P and Q to the x-axis to get a trapezoid, which is the triangle plus two right triangles that share the corner O. Then subtract and solve the resulting equation.

1STEP 1

Drop perpendiculars

Drop perpendiculars from P and Q to feet H and H' on the x-axis. As a ＞ 0, H is right of O and H' left, so O is on H'H.

H(a, 0), H'(-2a, 0)

2STEP 2

Area of trapezoid PQH'H

Its parallel sides are the vertical segments PH = b and QH' = 3b, and its height is H'H = a-(-2a) = 3a.

1/2 × (b + 3b) × 3a = 6ab

3STEP 3

Areas of the two right triangles

Triangle OQH' has legs OH' = 2a and QH' = 3b; triangle OPH has legs OH = a and PH = b.

△ OQH' = 1/2 × 2a × 3b = 3ab, △ OPH = 1/2 × a × b = ab/2

4STEP 4

Subtract to get triangle OPQ

Because O is on the bottom side, the trapezoid is exactly triangle OPQ plus the two right triangles.

6ab - (3ab + ab/2) = (5/2)ab

5STEP 5

Use the given area

Set the expression equal to 15 and solve for ab.

(5/2)ab = 15 → ab = 6

Answer

Try a=2, b=3 (so ab=6): P(2, 3), Q(-4, 9). The trapezoid is 1/2 × (3+9) × 6 = 36, minus 1/2 × 4 × 9 = 18 and 1/2 × 2 × 3 = 3, leaves 15.

Takeaway

Even with letters for coordinates, cut-and-subtract works -- and the answer can be a product you never split apart.

Drop perpendiculars
Area of trapezoid PQH'H
Areas of the two right triangles
Subtract to get triangle OPQ
Use the given area

Where next?

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

Box the triangle, subtract the corner triangles Triangle height depends on chosen base

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▶ Practice — 12 problems