# Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians

**Solution:**

As we know, if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. And we know that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

In ΔPQR, PM is the median and,

In ΔABC AN is the median

ΔPQR ∼ ΔABC (given)

∠PQR = ∠ABC................ (1)

∠QPR = ∠BAC.........(2)

∠QRP = ∠BCA...........(3)

and-PQAB = QR/BC = RP/CA............ (4)

(If two triangles are similar, then their corresponding angles are equal and corresponding sides are in the same ratio)

Area of ΔPQR / Area of ΔABC = (PQ)^{2} / ( AB)^{2} = (QR)²/(BC)² = (RP)^{2} / (CA)^{2}------ [THEROM 6.6] ……… (5)

Now In ΔPQM and ΔABN

∠PQM = ∠ABN...... (from 1)

And PQ/AB = QM/BN

[Therefore, PQ / AB = QR / BC = 2QM / 2BN; M, N midpoints of QR and BC]

⇒ ΔPQM ∼ ΔABN [SAS similarly]

Area of ΔPQM / Area of ΔABN = (PQ)^{2} / (AB)^{2} = (QM)^{2 }/ (BN)^{2} = (PM)^{2} / (AN)^{2} [by theorem 6.6]....... (6)

From (5) and (6)

Area of ΔPQR / Area of ΔABC = (PM)^{2} / ( AN)^{2}

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 6

**Video Solution:**

## Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians

NCERT Class 10 Maths Solutions Chapter 6 Exercise 6.4 Question 6

Hence proved that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians

**☛ Related Questions:**

- Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
- Tick the correct answer and justify:ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is(A) 2: 1(B) 1: 2(C) 4: 1(D) 1: 4
- Sides of two similar triangles are in the ratio 4: 9. Areas of these triangles are in the ratio(A) 2 : 3(B) 4: 9(C) 81: 16(D) 16: 81
- Let ∆ ABC ~ ∆ DEF and their areas be, respectively, 64 cm^2 and 121 cm^2. If EF = 15.4 cm, find BC.

visual curriculum