In the figure, we see three triangles within a rectangle. The areas of the triangles are A, B, C, as marked. Show that the area of the rectangle is 2(A + C)(B + C)C.
Step-by-step solution
Idea: As in the figure, the inner point R is directly below P and level with Q, so PR and RQ are parallel to the sides. Then each triangle has a vertical or horizontal side, and its area is easy to write.
- Put the bottom-left corner O at (0, 0); the rectangle is w wide and h high. From the figure: P = (p, h) on the top side, Q = (w, q) on the right side, and R = (p, q), so PR is vertical and RQ horizontal.1 mark
- C = △PRQ (right angle at R) = 12(h − q)(w − p). A = △OPR: base PR = h − q (vertical), height = p, so A = 12p(h − q). B = △ORQ: base RQ = w − p (horizontal), height = q, so B = 12q(w − p).1 mark
- A + C = 12(h − q)(p + w − p) = 12w(h − q). B + C = 12(w − p)(q + h − q) = 12h(w − p).1 mark
- 2(A + C)(B + C)C = [2 × ¼wh(h − q)(w − p)] ÷ [½(w − p)(h − q)] = wh = area of the rectangle. Shown.1 mark
Check: w = 10, h = 6, p = 6, q = 2: C = 12 × 4 × 4 = 8, A = 12 × 6 × 4 = 12, B = 12 × 2 × 4 = 4. 2 × 20 × 128 = 60 = 10 × 6 ✓.
Answer to write in the exam
O(0, 0), P(p, h), Q(w, q), R(p, q); rectangle = wh
C = 12(w − p)(h − q), A = 12p(h − q), B = 12q(w − p)
A + C = 12w(h − q), B + C = 12h(w − p)
2(A + C)(B + C) ÷ C = [12wh(h − q)(w − p)] ÷ [12(w − p)(h − q)]
∴ = wh = area of rectangle
Common mistakes that cost marks
- Using the slanting side OP as a base for A; use the vertical side PR with height p.
- Not stating that PR and RQ are parallel to the sides (read from the figure); the proof depends on it.
- Cancelling (w − p) with (h − q), which are different.
How this can come in the exam
In a figure like this one, A = 15, B = 9 and C = 5. The area of the rectangle is
- 29
- 56
- 80
- 112
Show answer
(D) 112
2 × 20 × 145 = 5605 = 112.
Try one yourself
In a figure like this one, A = 10, B = 6, C = 6. Find the area of the rectangle and the area outside the three triangles.
Show answer
2 × 16 × 126 = 64; outside = 64 − 22 = 42.
More questions like this
- In the figure we see two shaded regions formed by a quarter circle, a semicircle, and a triangle. Show that the areas of the two shaded regions are equal.
- In the figure, you see athletes assembled at the start of a 4 × 100 m relay race. The tracks are laid out, and the athletes are all set to go racing down the tracks. Do you notice that the athletes are not at the same starting line? Those in the outer lanes seem to be starting ahead of those in the inner lanes while the finish line is the same for all of them. What could be the reason for this? The distance between the starting points of adjacent lanes is called the ‘stagger’. Notice that the stagger continues all the way to the outermost lane. Do you think the stagger gives anyone (those in the outer lanes or in the inner lanes) an unfair advantage? Why or why not? On what basis can the organisers work out the length of the stagger between lanes?
- In my school, the playground is too small to have a 400 m track, so the school constructed a 200 m track instead. Does this mean that we need a smaller stagger for the race tracks in my school (i.e., smaller than the stagger used in the Olympics), for the same 4 × 100 m relay race?
- Here we see a circle with radius r units. What is its perimeter? How do we find out?
- What is the connection between this question and the one about the 400 m athletics track?