Question

Area of triangle ADE = 9 cm² and area of trapezium DBCE = 16 cm².

Statement (1): \[\frac{\mathrm{DE}}{\mathrm{BC}}=\frac{3}{4}\]

Statement (2): \[\frac{\triangle \mathrm{ADE}}{\triangle \mathrm{ABC}}=\frac{9}{25} \Rightarrow \frac{\mathrm{DE}}{\mathrm{BC}}=\frac{3}{5}\]

पर्याय

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Statement 1 is false, and statement 2 is true.

Explanation:

Given,

Area of triangle ADE = 9 cm² and area of trapezium DBCE = 16 cm².

In Δ ABC and Δ ADE,

⇒ ∠BAC = ∠DAE (Common angle)

⇒ ∠ABC = ∠ADE (Corresponding angles are equal)

⇒ ∠ACB = ∠AED (Corresponding angles are equal)

∴ Δ ABC ∼ Δ ADE (By A.A.A. postulate)

From figure,

⇒ Area of triangle ABC = Area of triangle ADE + Area of trapezium DBCE = 9 + 16 = 25 cm²

We know that,

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

∴ `("Area of"  Δ ADE)/("Area of"  Δ ABC​) = (DE^2)/(BC^2)​`

⇒ `9/25 ​= (DE^2)/(BC^2)​​`

⇒ `(DE)/(BC)​ = sqrt(9/25) ​​= 3/5​`

So, statement 1 is false but statement 2 is true.