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Area of triangles · point inside a rectangle
P6 · Area & perimeter · PSLE Maths
Question
Ali chose a point P inside rectangle ABCD and joined P to A, B, C and D. Rectangle ABCD is 24 cm long and 15 cm wide. Find the total area of triangle APB and triangle CPD.
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Hint
Think of APB and CPD as triangles with bases on the top and bottom of the rectangle. Their bases are equal, and their heights add up to the breadth of the rectangle.
Show the answer and step-by-step explanation
Answer: Total area 180 cm²
- Draw the figure — Point P is inside the rectangle. We need the total area of the top triangle APB and the bottom triangle CPD.
- Equal bases — The base of triangle APB is AB, and the base of triangle CPD is CD. Opposite sides of a rectangle are equal, so AB = CD = 24 cm.
- Two heights — Let the height from P to AB be h₁, and the height from P to CD be h₂. Since the top and bottom sides are parallel, h₁ + h₂ is exactly the breadth of the rectangle, 15 cm.
- Translate base and height — Translate the lower base CD up to the top. Its length stays the same, 24 cm. The two triangles act like they share the same base, and their two heights add up to 15 cm.
- Calculate area — Substitute h₁ + h₂ = 15 cm: total area = 1/2 × 24 × 15 = 180 cm². This is half the area of the rectangle.
- Key idea — 📌 Key idea: When a point inside a rectangle is joined to the four vertices, the two opposite triangles with bases on the top and bottom sides have equal bases, and their heights add up to the breadth. So their total area is half the area of the rectangle.
Method: When a point inside a rectangle is joined to the four vertices, the total area of the two opposite triangles with bases on the top and bottom sides is half the area of the rectangle. This is because the bases are equal, and the two heights add up to the breadth of the rectangle.
Practice: 3 similar questions
- Rectangle ABCD is 20 cm long and 12 cm wide. Point P is inside the rectangle and is joined to the four vertices. Find the total area of triangle APB and triangle CPD.
Answer
120 cm² — Area of rectangle = 20 × 12 = 240 cm². Total area of triangles APB and CPD = 240 ÷ 2 = 120 cm².
- Siti chose a point P inside a 30 cm × 18 cm rectangular poster ABCD and joined P to the four vertices. Find the total area of triangle BPC and triangle DPA.
Answer
270 cm² — Area of rectangle = 30 × 18 = 540 cm². Triangles BPC and DPA form the other pair of opposite triangles, so their total area is also half the area of the rectangle. Total area = 540 ÷ 2 = 270 cm².
- Rectangle ABCD is 28 cm long and 16 cm wide. Point P is inside the rectangle and is joined to the four vertices. The area of triangle APB is 70 cm². Find the area of triangle CPD.
Answer
154 cm² — Area of rectangle = 28 × 16 = 448 cm². Total area of triangles APB and CPD = 448 ÷ 2 = 224 cm². Area of triangle CPD = 224 − 70 = 154 cm².
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