If $A D$ and $P M$ are medians of triangles $A B C$ and $P Q R,$ respectively where $△ A B C \sim △ P Q R$ , prove that $\frac{A B}{P Q} = \frac{A D}{P M}$ .

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Solution

Since,
$△ A B C$ $\sim$ $△ P Q R$

$∴$ $\frac{A B}{P Q} = \frac{B C}{Q R} = \frac{A C}{P R}$

$∠ A = ∠ P, ∠ B = ∠ Q, ∠ C = ∠ R$

But, $B C = 2 B D$ and $Q R = 2 Q M$

Hence, $\frac{A B}{P Q} = \frac{B D}{Q M} = \frac{A C}{P R}$

In $△ A B D$ and $△ P Q M,$

$\frac{A B}{P Q} = \frac{B D}{Q M}$

and $∠ B = ∠ Q$

By SAS similarity,

$△ A B D$ $\sim$ $△ P Q M,$

$∴$ $\frac{A B}{P Q} = \frac{A D}{P M}$

Hence Proved

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Q. Question 16
If AD and PM are medians of triangles ABC and PQR, respectively where

Δ A B C \sim Δ P Q R

. Prove that

\frac{A B}{P Q} = \frac{A D}{P M}

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that : $\frac{A B}{P Q} = \frac{A D}{P M}$ .

△ A B C \sim △ P Q R

. If AD and PM are corresponding medians of the two triangles , then

\frac{A B}{P Q} = \frac{A D}{P M}

Q. In

△ A B C, A D

is the median to

B C

and in

△ P Q R

P M

is the median to

Q R

. If

\frac{A B}{P Q} = \frac{B C}{Q R} = \frac{A D}{P M}

. Prove that

△ A B C \sim △ P Q R

If AD and PM are medians of triangles ABC and PQR, respectively where

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If AD and PM are medians of triangles ABC and PQR, respectively where △ABC∼△PQR, prove that ABPQ=ADPM.

Since, △ABC ∼ △PQR∴ ABPQ=BCQR=ACPR∠A=∠P,∠B=∠Q,∠C=∠RBut, BC=2BD and QR=2QMHence, ABPQ=BDQM=ACPRIn △ABD and △PQM,ABPQ=BDQM and ∠B=∠QBy SAS similarity, △ABD ∼ △PQM,∴ ABPQ=ADPMHence Proved

If $A D$ and $P M$ are medians of triangles $A B C$ and $P Q R,$ respectively where $△ A B C \sim △ P Q R$ , prove that $\frac{A B}{P Q} = \frac{A D}{P M}$ .