The vertices of a Δ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that Calculate the area of theΔ ADE and compare it with the area of Δ ABC (Recall Theorem 6.2 and Theorem 6.6)

Question

The vertices of a &Delta; ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that Calculate the area of the&Delta; ADE and compare it with the area of &Delta; ABC (Recall Theorem 6.2 and Theorem 6.6)

Philoid · Accepted Answer

Given that,

= =

Therefore, D and E are two points on side AB and AC respectively such that they divide side AB and AC in a ratio of 1:3

Coordinates of point D = ()

= (, )

Coordinates of point E = (, )

= (, )

Area of the triangle = [x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ - y₂)

= [4 (– ) + ( - 6) + (6 – )]

= [3 – + ]

= []

= 15/32 square units

Area of triangle ABC = [4 (5 – 2) + 1 (2 - 6) + 7 (6 – 5)] /2

= (12 – 4 + 7)/ 2

= 15/2 square units

Clearly, the ratio between the areas of ΔADE and ΔABC is 1:16

We know that if a line segment in a triangle divides its two sides in the same ratio, then the line segment is parallel tothe third side of the triangle.

These two triangles so formed (here ΔADE and ΔABC) will be similar to each other.

Hence, the ratio between the areas of these two triangles will be the square of the ratio between the sides of these two triangles

And, ratio between the areas of ΔADE and ΔABC = (1/4)²

= 1/16