Measuring Space: Perimeter and Area | Exercise 6.2

Question 6

ABCD is a parallelogram. P and Q are any two points on side AB. What can you say about the ratio area (ΔPCD): area (ΔQCD)?

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Solution

We need to compare the areas of the two triangles.

Step 1 — Identify the common base

Let's look at triangle PCD. Its base is the side CD.

Now let's look at triangle QCD. Its base is also the side CD.

Both triangles share the same base. The common base is CD.

Diagram 1

Step 2 — Identify the common height

ABCD is a parallelogram. This means side AB is parallel to side CD.

Points P and Q lie on side AB. The perpendicular distance between two parallel lines is always the same. So, the perpendicular distance from point P to line CD is constant. The perpendicular distance from point Q to line CD is also constant.

This perpendicular distance is the height of triangle PCD with respect to base CD. This same distance is the height of triangle QCD with respect to base CD. Therefore, both triangles have the same height.

Step 3 — Compare the areas

The area of a triangle is given by the formula: Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

For triangle PCD: Area(ΔPCD)=12×CD×height\text{Area}(\Delta\text{PCD}) = \frac{1}{2} \times \text{CD} \times \text{height}

For triangle QCD: Area(ΔQCD)=12×CD×height\text{Area}(\Delta\text{QCD}) = \frac{1}{2} \times \text{CD} \times \text{height}

Since both triangles have the same base CD and the same height, their areas are equal. So, area(Δ\DeltaPCD) is equal to area(Δ\DeltaQCD).

The ratio of their areas is: area(ΔPCD):area(ΔQCD)=1:1\text{area}(\Delta\text{PCD}) : \text{area}(\Delta\text{QCD}) = 1 : 1

Answer

(i) The ratio area (Δ\DeltaPCD) : area (Δ\DeltaQCD) is 1 : 1.

More questions in Exercise 6.2

Q1

Find the area of triangle ADE in Fig. 6.31.

Q2

The parallel sides of a trapezium are 40 cm and 20 cm. If its non-parallel sides are both equal, each being 26 cm, find the area of the trapezium.

Q3

Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm.

Q4

The sides of a triangular plot are in the ratio 3: 5: 7; its perimeter is 300 m. Find its area.

Q5

One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area 128 cm², find the length of the shorter diagonal.

Q6

ABCD is a parallelogram. P and Q are any two points on side AB. What can you say about the ratio area (ΔPCD): area (ΔQCD)?

Q7

O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles PSO and PQO are equal.

Q8

If the mid-points of the sides of a 4-gon (also known as a quadrilateral, but we prefer to call it a ‘4-gon’) are joined in order, prove that the area of the parallelogram thus formed will be half of the area of the given 4-gon. (You may wonder whether the 4-gon thus formed is always a parallelogram, and if so, why? These questions will be tackled and answered in the chapter on quadrilaterals.)

Q9

In Δ\DeltaABC, the midpoint of BC is D (Fig. 6.32). Median AD is drawn. P is any point on AD. Show that area (Δ\DeltaABP) = area (Δ\DeltaACP).

Q10

Given a square ABCD, let P be a point within it. Join PA, PB, PC, PD (Fig. 6.33). What is the ratio of the areas of the red region (Δ\DeltaPAB and Δ\DeltaPCD) and the green region (Δ\DeltaPBC and Δ\DeltaPDA)?

Q11

In Δ\DeltaABC, D is the midpoint of AB. P is any point on BC, and Q is a point on AB such that CQ || PD. PQ is joined (Fig. 6.34). Prove that

Area (ΔBPQ)=12Area (ΔABC)\text{Area }(\Delta\text{BPQ}) = \frac{1}{2} \text{Area }(\Delta\text{ABC})

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