Name: Areas of Similar Triangles
Uploaded: 2019-07-19
Duration: 6 min 54 s
Description: The corresponding sides of similar triangles are proportional. Are the ratios of areas of similar triangles the same as that of their sides? Watch this video...

Question

Sides $A B$ and $A C$ and median $A D$ of a triangle $A B C$ are respectively proportional to sides $P Q$ and $P R$ and median $P M$ of another triangle $P Q R$ . Show that $∆ A B C ~ ∆ P Q R$ .

Accepted Answer

Given that,
$A C$

Let us extend $A D$ and $P M$ up to point $E$ and $L$ respectively such that $A D = D E$ and $P M = M L$ .

Now join $B$ to $E$ , $C$ to $E$ , $Q$ to $L$ and $R$ to $L$ .

We know that medians divide opposite sides.

So, $B D = D C$ and $Q M = M R$

Also, $A D = D E$ (by construction)

And $P M = M L$ (By construction)

So, in quadrilateral $A B E C$ , diagonals $A E a n d B C$ bisects each other at point $D$ .

Also, in quadrilateral $P Q$ , diagonals $P L a n d Q R$ bisects each other at point $M$ .

So, quadrilaterals $A BED$ and $P Q L R$ are parallelograms.

$A C = B E$ and $A B = E C$ (Since it is a parallelogram, opposite sides will be equal)

Also $P R = Q L$ and $P Q = L R$ (Since it is a parallelogram, opposite sides will be equal)

In $∆ A B E a n d ∆ P Q L$ ,

$\frac{A B}{P Q} = \frac{B E}{Q L} = \frac{A E}{P L}$ $(\frac{A C}{P R} = \frac{B E}{Q L} a n d \frac{A D}{P M} = \frac{2 A D}{2 P M} = \frac{A E}{P L})$

Hence, by $S S S$ rule,

$∆ A B E ~ ∆ P Q L$

Similarly, $∆ A E C ~ ∆ P L R$

Hence, $∠ B A E = ∠ Q P L$ and $∠ E A C = ∠ L P R$

Hence, $∠ B A C = ∠ Q P R$

Now, in $∆ A B C a n d ∆ P Q R$ ,

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

Hence, by $S A S$ rule,

$∆ A B C ~ ∆ P Q R$

Sides $A B$ and $A C$ and median $A D$ of a triangle $A B C$ are respectively proportional to sides $P Q$ and $P R$ and median $P M$ of another triangle $P Q R$ . Show that $∆ A B C ~ ∆ P Q R$ .

Important Questions on Triangles

Learn and Prepare for any exam you want !

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 ∆ABC~∆PQR.

Important Questions on Triangles

Learn and Prepare for any exam you want !

Sides $A B$ and $A C$ and median $A D$ of a triangle $A B C$ are respectively proportional to sides $P Q$ and $P R$ and median $P M$ of another triangle $P Q R$ . Show that $∆ A B C ~ ∆ P Q R$ .