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State if the following is true or false.

P and Q are points on the sides AB and AC respectively of a ΔABC. If AP = 2 cm, PB = 4 cm, AQ = 3 cm and QC = 6 cm, then BC = 3 PQ.

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True

False

answer is A.

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Detailed Solution

Given,
P and Q are points on the sides AB and AC respectively of a ΔABC, AP = 2 cm, PB = 4 cm, AQ = 3 cm and QC = 6 cm.
Question Image

The ratio of the corresponding sides of the triangles are,

\frac{AP}{PB} = \frac{2}{4}

\Rightarrow \frac{AP}{PB} = \frac{1}{2}

…….(1)

\frac{AQ}{QC} = \frac{3}{6}

\Rightarrow \frac{AQ}{QC} = \frac{1}{2}

…..(2)
So, from equation (1) and (2),
⇒

A P P B = A Q Q C

⇒

P Q ∥ B C

Consider the triangles

Δ A P Q

and

Δ A B C

∠ A P Q

and

∠ A B C

are corresponding angles.
⇒

∠ A P Q = ∠ A B C

∠ A Q P

and

∠ A C B

are corresponding angles.
⇒

∠ A Q P = ∠ A C B

.
Therefore, the triangles

Δ A P Q

and

Δ A B C

are similar.
As the triangles are similar.
So, the corresponding sides will have the same ratio,

A P A B = P Q B C ⇒ 26 = P Q B C ⇒ 13 = P Q B C ⇒ B C = 3 P Q

Therefore, the statement is true.
Hence, option 1 is correct.

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