Question

Which of the following is a false statement?

Options

• If the areas of two similar triangles are equal then the triangles are congruent.

• The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides.

• The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding medians.

• The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding altitudes.

Answer 1

The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides.

Explanation:

a. Let AE = x cm.

Then, EC = (5.6 – x) cm.

`(AD)/(DB) = (AE)/(EC)` ⇒ `3/5 = x/(5.6 - x)`

∴ 3(5.6 – x) = 5x

⇒ 8x = 3 × 5.6

∴ `x = (3 xx 5.6)/8`

= `(16.8)/8`

= 2.1

b. `(AB)/(DE) = (BC)/(EF)`

⇒ `3/2 = 6/x`

⇒ 3x = 12

⇒ x = 4

c. `(ar(ΔABC))/(ar(ΔPQR)) = (BC^2)/(QR^2)`

⇒ `9/16 = (BC^2)/(QR^2)`

⇒ `(3/4)^2 = ((BC)/(QR))^2`

⇒ `(BC)/(QR) = 3/4`

⇒ `QR = 4/3 xx BC`

= `(4/3 xx 4.5) cm` 

= 6 cm

d. ΔОАВ ~ ΔОCD

⇒ `(OA)/(OC) = (OB)/(OD)`

⇒ `(2x + 4)/(2x - 1) = (9x - 21)/3`

⇒ 6x + 12 = 18x² – 51x + 21

⇒ 18x² – 57x + 9 = 0

⇒ 6x² – 19x + 3 = 0

⇒ (x – 3)(6x – 1) = 0

⇒ x = 3 or x = `1/6`

But, `x = 1/6` makes (2x – 1) < 0. So, we reject it.

∴ x = 3.