Choose whether the given statement is true or false.

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Then ΔABC ∼ ΔPQR.

Question

Choose whether the given statement is true or false.

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Then ΔABC ∼ ΔPQR.

Accepted Answer

Let us consider that AB and PQ are proportional and AC and PR are proportional from ΔABC, ΔPQRΔ.
Then,

\frac{AB}{PQ}

=

\frac{AC}{PR}

=

\frac{AD}{PM}

............…. (1)
Now let us produce AD to E, such that AD = DE. Similarly, produce PM to N such that PM = MN.
Now join BE, CE, QN, RN as shown in the figure.

https://www.vedantu.com/question-sets/644a8bbf-00a8-449c-96ee-e12b95cfbd13398534452219190843.png

Thus, we get a quadrilateral ABEC and PQNR. They are parallelograms because their diagonals bisect each other at point D and M.
Now consider parallelogram ABEC and PQNR BE = AC and QN = PR, opposite sides of the parallelogram are equal. Hence, we can write it as,
⇒

\frac{BE}{AC}

= 1,

\frac{QN}{PR}

= 1
Now from the above, we can say that,
⇒

\frac{BE}{AC}

=

\frac{QN}{PR}

Cross-multiply them,
⇒

\frac{AC}{PR}

=

\frac{BE}{QN}

Compare with equation (1),
∴

\frac{AB}{PR}

=

\frac{BE}{QN}

....................... (2)
From equation (1),

\frac{AB}{PQ}

=

\frac{AD}{PM}

Multiply

\frac{AD}{PM}

by 2,
⇒

\frac{AB}{PQ}

=

\frac{2 AD}{2 PM}

Substitute the values,
⇒

\frac{AB}{PQ}

=

\frac{AE}{PN}

..................….. (3)
From equation (2) and (3),
∴

\frac{BE}{QN}

=

\frac{AE}{PN}

By SSS similarity,
ΔABE ∼ ΔPQN
So, the corresponding angles of 2 triangles are equal.
⇒ ∠BAE = ∠QPN............….. (4)
Similarly, ΔACE ∼ ΔPRN.
So, the corresponding angles of 2 triangles are equal.
⇒ ∠CAE = ∠RPN ..........….. (5)
Add the equations (4) and (5),
⇒ ∠BAE + ∠CAE = ∠QPN + ∠RPN
From the figure,
⇒ ∠BAC = ∠BAE + ∠CAE, ∠QPR = ∠QPN + ∠RPN Substitute the values,
∴ ∠BAC = ∠QPR .............….. (6)
Now, in ΔABC and ΔPQR,
⇒

\frac{AB}{PQ}

=

\frac{AC}{PR}

(from (1))
⇒ ∠BAC = ∠QPR (from (6))
By SAS similarity,
∴ ΔABC ∼ ΔPQR
Hence, it is proved.
So, the given statement is true.

Choose whether the given statement is true or false.

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Then ΔABC ∼ ΔPQR.

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