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Q.

Two triangles ABC and PQR are similar, if BC : CA : AB = 1 : 2 : 3, then $\frac{QR}{PR}$ is _______.

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a

\frac{1}{3}

b

\frac{1}{2}

c

\frac{1}{\sqrt{2}}

d

\frac{2}{3}

answer is B.

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Detailed Solution

Triangles are said to be similar if they have the same shape, but same sizes may not be necessary.
Here the given two triangles ABC and PQR are similar.
ΔABC ∼ ΔPQR
Properties of similar triangles are like corresponding angles will be the same and corresponding sides will be in the same proportions.
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and

\frac{AB}{PQ}

=

\frac{BC}{QR}

=

\frac{AC}{PR}

.

1454(1).jfif

1454(2).jfif

Here for triangle ABC, it is given that BC : CA : AB = 1 : 2 : 3
It is represented in ratio form as

\frac{BC}{1}

=

\frac{CA}{2}

=

\frac{AB}{3}

. Assuming constant one side BC = k, so the equation becomes

\frac{k}{1}

=

\frac{CA}{2}

=

\frac{AB}{3}

.

\frac{k}{1}

=

\frac{CA}{2}

. So, CA = 2k.
And

\frac{k}{1}

=

\frac{AB}{3}

. So, AB = 3k
As Triangle ABC and PQR are similar triangle,

\frac{AB}{PQ}

=

\frac{BC}{QR}

=

\frac{AC}{PR}

So,

\frac{BC}{QR}

=

\frac{AC}{PR}

Putting value BC = k and CA = 2k in above equation,

\frac{k}{QR}

=

\frac{2 k}{PR}

Simplifying,

\frac{k}{2 k}

=

\frac{QR}{PR}

,
So,

\frac{1}{2}

=

\frac{QR}{PR}

.
So,

\frac{QR}{PR}

=

\frac{1}{2}

So, Option (2) is the correct answer.

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