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Home » Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.
Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.
CBSE | Class 10 | Coordinate Geometry | Maths | NCERT
Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

Solution:

Let the triangle’s vertices be A (0, -1), B (2, 1), and C (3, 1). (0, 3).

Let D, E, and F be the midpoints of the triangle’s sides.

D, E, and F coordinates are given by

D .= (0+2/2, -1+1/2 ) = (1, 0)

E = ( 0+0/2, -1+3/2 ) = (0, 1)

F = ( 0+2/2, 3+1/2 ) = (1, 2)

The area of a triangle

    \[=~1/2\text{ }\times \text{ }[{{x}_{1}}({{y}_{2}}~\text{ }{{y}_{3}})\text{ }+\text{ }{{x}_{2}}({{y}_{3}}~\text{ }{{y}_{1}})\text{ }+\text{ }{{x}_{3}}({{y}_{1}}~\text{ }{{y}_{2}})]\]

The area of ΔDEF

    \[=\text{ }1/2\text{ }\left\{ 1\left( 2-1 \right)\text{ }+\text{ }1\left( 1-0 \right)\text{ }+\text{ }0\left( 0-2 \right) \right\}\text{ }=\text{ }1/2\text{ }\left( 1+1 \right)\text{ }=\text{ }1\]

Area of ΔDEF is 1 square units

The area of ΔABC

    \[=\text{ }1/2\text{ }\left[ 0\left( 1-3 \right)\text{ }+\text{ }2\left\{ 3-\left( -1 \right) \right\}\text{ }+\text{ }0\left( -1-1 \right) \right]\text{ }=\text{ }1/2\text{ }\left\{ 8 \right\}\text{ }=\text{ }4\]

Area of ΔABC is 4 square units

As a result, 1:4 is the required ratio.

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