of A ABC16. If D is a point on the side AB of A ABC such that AD: DB -

1.	of A ABC16. If D is a point on the side AB of A ABC such that AD: DB -3.2 and E is a point on BC such[CBSE 2006CL nndARnF are rqvilateral triangles, where D is the mid point of BC, find thethat DE 11 A C. Find the ratio of areas of Î ABC and Î BDE.
Answer» Given : In ΔABC, D is appoint on the side AB such that AD:DB = 3:2. E is a point on side BC such that DE \|\| AC. Let AD = 2x and BD = 3x In ΔABC and ΔBDE, ∠BDE =∠A (corresponding angles) ∠DBE =∠ABC (common) ΔABC∼ΔBDE [By AA similarity] We know that the ratio of the two similar triangles is equal to the ratio of the squares of their corresponding sides arΔABC/ arΔBDE = (AB/BD)² arΔABC/ arΔBDE = ((BD+DA) /BD)² [AB = BD + DA] arΔABC/ arΔBDE = ((3x+2x) /2x)² arΔABC/ arΔBDE = (5x / 2x)² arΔABC/ arΔBDE = 25x / 4x arΔABC/ arΔBDE = 25/4 Hence, the ratio of areas of ΔABC and ΔBDE is 25 : 4.

of A ABC16. If D is a point on the side AB of A ABC such that AD: DB -3.2 and E is a point on BC such[CBSE 2006CL nndARnF are rqvilateral triangles, where D is the mid point of BC, find thethat DE 11 A C. Find the ratio of areas of Î ABC and Î BDE.

Answer»

Given : In ΔABC, D is appoint on the side AB such that AD:DB = 3:2. E is a point on side BC such that DE || AC.

Let AD = 2x and BD = 3x

In ΔABC and ΔBDE,

∠BDE =∠A (corresponding angles)

∠DBE =∠ABC (common)

ΔABC∼ΔBDE [By AA similarity]

We know that the ratio of the two similar triangles is equal to the ratio of the squares of their corresponding sides

arΔABC/ arΔBDE = (AB/BD)²

arΔABC/ arΔBDE = ((BD+DA) /BD)²

[AB = BD + DA]

arΔABC/ arΔBDE = ((3x+2x) /2x)²

arΔABC/ arΔBDE = (5x / 2x)²

arΔABC/ arΔBDE = 25x / 4x

arΔABC/ arΔBDE = 25/4

Hence, the ratio of areas of ΔABC and ΔBDE is 25 : 4.