Question

find the area of triangle formed by joining the midpoints of the two 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.

Answer 1

◀ HEY THERE!! ◀

◀ Question: ◀

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.

◀ Method of Solution:◀

→ Let the vertices of the triangle be A (0, -1), B (2, 1), C (0, 3).

→ Let D, E, F be the midpoints of the sides of this triangle. Coordinates of D, E, and F are given,

→ Step:(1) Coordinate of D

⇒ D=(0+2/2 , -1+1/2) = 1,0

→ Step : (2) Coordinate of E

⇒ E =(0+0/2 , 3-1/2) = 0,1

→ Step:(3) Coordinate of F

⇒ F= (2+0/2 , 1+3/2) = 1,2

Now,

→ Area of triangle(∆DEF) = (1/2)[x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)] →

⇒ Area of Triangle =(1/2)[1(2-1) + 1(1-0) + 0(0-2)]

⇒ Area of Triangle = 1/2(1+1) => 1 units

Again,

→ Area of triangle(∆ABC) = (1/2)[x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)]. →

•°• Area of ∆ABC = (1/2)[0(1-3) + 2{3-(-1)} + 0(-1-1)}]

⇒ Area of Triangle (∆ABC ) = 1/2 × 8

•°• Area of (∆ABC) = 4 units

◀ Hence, Ratio of ∆DEF and ∆ABC is 1:4 ◀

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Answer 2

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

Let x, y, z be the midpoints of the sides of this triangle. Coordinates of x, y, and z are given by

x=(0+2/2 , -1+1/2) = 1,0

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

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

Area of Triangle =(1/2)[1(2-1) + 1(1-0) + 0(0-2)]

Area of Triangle = 1/2(1+1)
= 1

Area of ∆ABC = (1/2)[0(1-3) + 2[3-(-1)] + 0(-1-1)]

Area of Triangle ∆ABC = 1/2 × 8

Area of ∆ABC = 4

So, Ratio is 1:4.