What fraction of the triangle is shaded? What fraction of the square is shaded?
- (a) What fraction of the triangle is shaded?
- (b) What fraction of the square is shaded?
Step-by-step solution
Idea: (a) Compare triangles that share a vertex: their areas are in the ratio of their bases on a common line. (b) Put the square on a grid and find the corners of the small square.
(a) What fraction of the triangle is shaded?
- Name the triangle PQR (P at the top, Q bottom-left, R bottom-right). M is the midpoint of PQ, and X, Y cut PR into three equal parts (PX = XY = YR). The shaded region is QMXY. Let ar(△PQR) = S.½ mark
- △PQY and △PQR have the same height from Q and bases PY = 23PR on the same line, so ar(△PQY) = 23S. Likewise ar(△PQX) = 13S, and XM is a median of △PQX, so ar(△PMX) = 12 × 13S = 16S.1 mark
- Shaded = ar(△PQY) − ar(△PMX) = 23S − 16S = 12S. Half the triangle is shaded.½ mark
(b) What fraction of the square is shaded?
- Place the square with corners (0, 0), (2, 0), (2, 2), (0, 2) (side 2, area 4). The four lines are: from (1, 2) to (2, 0): y = −2x + 4; from (0, 2) to (1, 0): y = −2x + 2; from (0, 1) to (2, 2): y = x2 + 1; from (0, 0) to (2, 1): y = x2.½ mark
- Solving pairs of these gives the corners of the small square: (65, 85), (25, 65), (45, 25), (85, 45). For example, −2x + 4 = x2 + 1 gives x = 65, y = 85.1 mark
- Side2 = (65 − 25)2 + (85 − 65)2 = 1625 + 425 = 45. So the shaded area is 45 out of 4: 15 of the square.½ mark
Check: (a) With P(0, 6), Q(0, 0), R(6, 0): M(0, 3), X(2, 4), Y(4, 2); shoelace area of QMXY = 9 = 12 × 18 ✓. (b) 5 × 45 = 4, the area of the big square ✓.
Answer to write in the exam
(a)
ar(△PQY) = 23ar(△PQR) (PY = 23PR, same height)
ar(△PQX) = 13ar(△PQR); ar(△PMX) = 12ar(△PQX) = 16ar(△PQR) (XM median)
Shaded = 23 − 16 = 12
∴ 12 of the triangle is shaded.
(b)
Side 2: lines y = −2x + 4, y = −2x + 2, y = x2 + 1, y = x2
Corners of small square: (65, 85), (25, 65), (45, 25), (85, 45)
Side2 = (45)2 + (25)2 = 45
∴ Fraction = 45 ÷ 4 = 15
Common mistakes that cost marks
- In (a), thinking the shaded part is 13 because one side is cut into thirds.
- In (a), forgetting to subtract the small top triangle PMX.
- In (b), guessing 14 because there are 4 lines; the small square is 15.
How this can come in the exam
In △PQR, X is the point on PR with PX = 13PR. The ratio ar(△PQX) : ar(△XQR) is
- 1 : 1
- 1 : 2
- 1 : 3
- 2 : 3
Show answer
(B) 1 : 2
Same height from Q; bases PX : XR = 1 : 2.
Try one yourself
In a triangle, the midpoints of two sides are joined. What fraction of the triangle lies above this line (towards the shared vertex)?
Show answer
14 (the small triangle has half the base and half the height).
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