In given figures sides $A B$ and $B C$ and median $A D$ of a $△ A B C$ are respectively proportional to sides $P Q, Q R$ and median $P M$ of $△ P Q R$ . Show that $△ A B C \sim △ P Q R$ .

Open in App

Solution

Given $A D$ and $P M$ are medians of $△ A B C$ and $△ P Q R,$

$∴$ $B D = \frac{1}{2} B C$ and $Q M = \frac{1}{2} Q R . . . . . . . . (1)$
Given that,
$\frac{A B}{P Q} = \frac{B C}{Q R} = \frac{A D}{P M} . . . . . . . . . (2)$
$∴$ From (1) and (2),
$\frac{A B}{P Q} = \frac{B D}{Q M} = \frac{A D}{P M} . . . . . . . . . . . (3)$
In $△ A B D$ and $△ P Q M$

$\frac{A B}{P Q} = \frac{B D}{Q M} = \frac{A D}{P M}$ [From(3)]

$∴$ By SSS criterian of proportionality $△ A B D$ $\sim$ $△ P Q M$

$∴$ $∠ B = ∠ Q$ (Corresponding Sides of Similar Triangles) $. . . . . . . . . . . . . (4)$
In $△ A B C$ and $△ P Q R$

$\frac{A B}{P Q} = \frac{B C}{Q R}$ (From 2)

$∠ B = ∠ Q$ (From 4)

$∴$ By SAS criterian of proportionality $△ A B C$ $\sim$ $△ P Q R$ $[h e n c e p r o v e d]$

Suggest Corrections

Similar questions

Q. Side

A B

and

B C

and median

A D

of a triangle

A B C

are respectively proportional to sides

P Q

and

Q R

and median

P M

P Q R

. Show that

A B C \sim P Q R

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR.

Q. Question 12
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that

Δ A B C \sim Δ P Q R

Q. Question 14
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. Show that

Δ A B C \sim Δ P Q R

Q. Sides

A B, A C

and median

A D

of a triangle

A B C

are respectively proportional to sides

P Q, P R

and median

P M

of another triangle

P Q R

. Show that

△ A B C \sim △ P Q R

Join BYJU'S Learning Program

In given figures sides AB and BC and median AD of a △ABC are respectively proportional to sides PQ,QR and median PM of △PQR. Show that △ABC∼△PQR.

In given figures sides $A B$ and $B C$ and median $A D$ of a $△ A B C$ are respectively proportional to sides $P Q, Q R$ and median $P M$ of $△ P Q R$ . Show that $△ A B C \sim △ P Q R$ .