Question

Find the area of ABC with vertices A(0, -1), B(2, 1) and C(0, 3). Also. find the area of the triangle formed by joining the midpoints of its sides .Show that the ratio of the areas of two triangles is 4: 1.

Answer 1

Step-by-step explanation:

• this is your answer firstly find the area of triangle after that find midpoint of the triangle and join midpoint and after joining find area of new triangle made by joining maidpoint

Answer 2

Area of triangle ABC=4 square units

The area of triangle PQR=1 square units

Step-by-step explanation:

The vertices of triangle ABC are

A(0,-1),B(2,1) and C(0,3)

$x_1=0,x_2=2,x_3=0,y_1=-1,y_2=1,y_3=3$

Area of triangle=$\frac{1}{2}\mid{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)}\mid$

Using the formula

Area of triangle ABC=$\frac{1}{2}\mid{0(1-3)+2(3+1)+0(-1-1)}\mid$

Area of triangle ABC=$\frac{1}{2}\mid 8\mid=4$ square units

Let P,Q and R mid points of side AB, BC and AC

Mid-point formula:$x=\frac{x_1+x_2}{2},y=\frac{y_1+y_2}{2}$

Using mid-point formula

The coordinates of P

$x=\frac{0+2}{2}=1,y=\frac{-1+1}{2}=0$

The coordinates of P are (1,0)

The coordinates of Q

$x=\frac{2+0}{2}=1,y=\frac{1+3}{2}=\frac{4}{2}=2$

The coordinates of Q are (1,2)

The coordinates of R are

$x=\frac{0+0}{2}=0,y=\frac{-1+3}{2}=\frac{2}{2}=1$

The coordinates of R are (0,1)

$x_1=1,x_2=1,x_3=0,y_1=0,y_2=2,y_3=1$

The area of triangle PQR

=$\frac{1}{2}\mid{1(2-1)+1(1-0)+0(0-2)}\mid$

The area of triangle PQR=$\frac{1}{2}\mid{1+1+0}\mid=\frac{1}{2}\times 2=1$ square units

Ratio of area of triangle ABC to the area of triangle PQR=$\frac{4}{1}$

Hence,the ratio of the areas of two triangles 4:1.

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