If - ABC and - PQR are two similar triangles shown in the figure- AM and PN are the medians on - ABC and - PQR respectively- Then which of the following statements is true-A- - ABM - PQRB- - AMC - PNRC- - ABM - PQND- - ABM - PNQ

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Relation between Area and Sides of Similar Triangles

If Δ ABC and ...

Question

If $Δ$ ABC and $Δ$ PQR are two similar triangles shown in the figure. AM and PN are the medians on $Δ$ ABC and $Δ$ PQR respectively. Then which of the following statements is true?

Δ A B M \sim Δ P Q R

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Δ A M C \sim Δ P N R

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Δ A B M \sim Δ P Q N

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Δ A B M \sim Δ P N Q

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Solution

The correct options are
B $Δ A M C \sim Δ P N R$
C $Δ A B M \sim Δ P Q N$
Since $Δ$ ABC and $Δ$ PQR are similar, ratio of corresponding sides is same
$\Rightarrow \frac{A B}{P Q} = \frac{B C}{Q R} = \frac{C A}{R P}$
Medians AM and PN divide BC and QR into two equal parts.
$\Rightarrow B M = \frac{B C}{2}, Q N = \frac{P N}{2}$
In $Δ$ ABM and $Δ$ PQN,
we know that $\frac{A B}{P Q} = \frac{B C}{Q R}$
$\Rightarrow \frac{A B}{P Q} = \frac{2 B M}{2 Q N}$
$\Rightarrow \frac{A B}{P Q} = \frac{B M}{Q N}$ . (Since $∠ B = ∠ Q$ )
According to SAS criteria of similarity, $Δ$ ABM and $Δ$ PQN are similar.
Similarly, $Δ$ AMC and $Δ$ PNR are similar triangles.

Suggest Corrections

Similar questions

Q. Two sides AB, BC and median AM of one triangle ABC are respectively.
equal to sides PQ and QR and median PN of ΔPQR (See figure)
(i) ΔABM ≅ ΔPQN
(ii) ΔABC ≅ ΔPQR

Q. Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure).
Is

Δ A B M ≅ Δ P Q N

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that:

(i) ΔABM ≅ ΔPQN

(ii) ΔABC ≅ ΔPQR

Δ A B C

and

Δ P Q R

Δ A B C

and

Δ P Q R

, respectively. The ratio of areas of

Δ A B C

and

Δ P Q R

is 9:25. If AM = PO = 5 cm. Find the value of 3(ON).

Q. Two sides

A B

and

B C

and median

A M

of one triangle

A B C

are respectively equal to side

P Q

and

Q R

and median

P N

Δ P Q R

(see figure). Show that
(i)

Δ A B M = Δ P Q N

(ii)

Δ A B C = Δ P Q R

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If ΔABC and ΔPQR are two similar triangles shown in the figure. AM and PN are the medians on ΔABC and Δ PQR respectively. Then which of the following statements is true?

If $Δ$ ABC and $Δ$ PQR are two similar triangles shown in the figure. AM and PN are the medians on $Δ$ ABC and $Δ$ PQR respectively. Then which of the following statements is true?