Question

PQ || BC. The ratio of the perimeter of triangle ABC to the perimeter of triangle APQ is 3:1. Given that the numerical value of the area of triangle APQ is a whole number, which of the following could be the area of the triangle ABC? 28,60,99 or 120​

Answer 1

Given:

ΔABC where PQ||BC

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

To find:

Area of ΔABC.

Solution:

Consider the triangles ABC and APQ where PQ||BC.

When two lines are parallel, and a transversal cuts through these parallel lines, the corresponding angles are equal. Here, we have parallel lines PQ and BC with transversals AC and AB on both sides as shown in figure.

Here, from ΔABC and ΔAPQ,

∠APQ = ∠ABC (corresponding angles)

∠AQP = ∠ACB (corresponding angles)

∠A = ∠A (common angle)

Hence, by AAA similarity criterion ΔABC ≈ ΔAPQ.

Thus, their corresponding sides are in same ratio.

The perimeters of the triangles are in the ratio $3:1$.

If the ratio of perimeters of two similar triangles have a scale factor of $a:b$, then the ratio of areas of the two triangles have a scale factor of $a^{2} :b^{2}$.

Let $P_{1}$ and $P_{2}$ be the perimeters of ΔABC and ΔAPQ respectively.

Let $A_{1}$ and $A_{2}$ be the areas of ΔABC and ΔAPQ respectively.

Then,

$\frac{A_{1}}{A_{2}} =\frac{P_{1}^{2}}{P_{2}^{2}}$

$\frac{A_{1}}{A_{2}}=\frac{3^{2}}{1^{2}}$

$\frac{A_{1}}{A_{2}}=\frac{9}{1}$

$A_{1}=9A_{2}$

Here, we can see that $A_{1}$ is a multiple of $9$.

Out of the four options given, $28,60,99,120$, only $99$ is divisible by $9$.

Hence, $A_{2}$ should be equal to $11sq.unit$ for $A_{1}$ to be $99sq.units$

∴ $A_{1}=9*11=99sq.units$

$A_{2}=11sq.units$

Hence, option (c) is correct.

Area of triangle ABC is $99sq.units$ and option (c) is the correct answer.

Answer 2

Answer:

Consider the triangles ABC and APQ where PQ||BC.

When two lines are parallel, and a transversal cuts through these parallel lines, the corresponding angles are equal. Here, we have parallel lines PQ and BC with transversals AC and AB on both sides as shown in figure.

Here, from ΔABC and ΔAPQ,

∠APQ = ∠ABC (corresponding angles)

∠AQP = ∠ACB (corresponding angles)

∠A = ∠A (common angle)

Hence, by AAA similarity criterion ΔABC ≈ ΔAPQ.

Thus, their corresponding sides are in same ratio.

The perimeters of the triangles are in the ratio 3:13:1 .

If the ratio of perimeters of two similar triangles have a scale factor of a:ba:b , then the ratio of areas of the two triangles have a scale factor of a^{2} :b^{2}a2:b2 .

Let P_{1}P1 and P_{2}P2 be the perimeters of ΔABC and ΔAPQ respectively.

Let A_{1}A1 and A_{2}A2 be the areas of ΔABC and ΔAPQ respectively.

Then,

\frac{A_{1}}{A_{2}} =\frac{P_{1}^{2}}{P_{2}^{2}}A2A1=P22P12

\frac{A_{1}}{A_{2}}=\frac{3^{2}}{1^{2}}A2A1=1232

\frac{A_{1}}{A_{2}}=\frac{9}{1}A2A1=19

A_{1}=9A_{2}A1=9A2

Here, we can see that A_{1}A1 is a multiple of 99 .

Out of the four options given, 28,60,99,12028,60,99,120 , only 9999 is divisible by 99 .

Hence, A_{2}A2 should be equal to 11sq.unit11sq.unit for A_{1}A1 to be 99sq.units99sq.units

∴ A_{1}=9*11=99sq.unitsA1=9∗11=99sq.units

A_{2}=11sq.unitsA2=11sq.units

Hence, option (c) is correct.

Area of triangle ABC is 99sq.units99sq.units and option (c) is the correct answer.

Step-by-step explanation:

the correct answer is (c)