Question

The ratio of the perimeter of triangle ABC to the perimeter of triangle APQ is 3/1 . Given thatthe numerical value of the area of triangle APQ is a whole number, which of the followingcould be the area of the triangle ABC?
(b) 60 (a) 28(c) 99(d) 1 20​

Answer 1

Answer:

If a line is drawn parallel to one side of a triangle to intersect the

other two sides in distinct points, the other two sides are divided in the same ratio.

As PQ∥BC

So

PB

AP

=

QC

AQ

∠AQP=∠ACB

∠APQ=∠ABC

So by AAA △AQP∼△ACB

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Hence

Area(ABC)

Area(APQ)

=

(AB)

2

(AP)

2

Area(ABC)

Area(APQ)

=

(AP+PB)

2

(AP)

2

Area(ABC)

Area(APQ)

=

(3x)

2

(x)

2

Area(ABC)

Area(APQ)

=

9

1

Let Area(APQ)=k

Area(ABC)=9k

Area(BPQC)=Area(ABC)−Area(APQ)=9k−k=8k

Area(BPQC)

Area(APQ)

=

8

1

∴ the ratio of the △APQ and trapezium BPQC =

8

1

Answer 2

Triangle

(c) 99

Ratio of perimeter of ΔABC to perimeter of ΔAPQ $= 3:1$

let the triangles ABC and APQ ware constructed in such a way that $PQ||BC$.

And we know that when two lines are parallel, and a transversal cuts through these parallel lines, the corresponding angles are equal.

Here PQ and BC are parallel lines  and AB and AC are transversal.

In $\triangle ABC$ and $\triangle APQ$,

$\angle APQ = \angle ABC$ (corresponding angles  are equal)

$\angle AQP = \angle ACB$ (corresponding angles are equal)

$\angle A = \angle A$ (common angle in both triangle)

So, $\triangle ABC \ and \ \triangle APQ \ are \ similar\ by \ AAA \ similarity\ criterion.$

$\implies \triangle ABC \sim\triangle APQ$

hence,  their corresponding sides are in same ratio and it is given that the perimeters of the both the triangles are in the ratio of 3:1 .

And it is known that the ratio of area of two triangle will be same as the square of the perimeters of both the triangles.

Let the area of triangle ABC is A1 and that of APQ is A2. So,

$\frac{A1}{A2}=\frac{3^2}{1^2}$

$\implies A1=9A2$

as we can see that area of ABC is multiple of 9, and from the option given we can see that only option (c) is the multiple of 9, given that area of APQ is whole number.

Hence, the area will be 99 sq. unit.