Question

Find ratio of triangle formed by joining the points (0,-1),(2,1),(0,3) and the area of triangle formed by joining the mid points of sides of triangle

Answer 1

Answer:

Ratio can't be fine

Step-by-step explanation:

The ratio of first triangle and newly formed triangle by midpoints of sides of a triangle is always 1 : 4, because first triangle is always four times of second triangle.

The Area of a triangle can be determined by:

$Area=\frac{x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1}+x_{3}(y_{1}-y_{2}}{2}$  

The Mid-points (x, y) of a triangle can be determined by:

$x=\frac{x_{1}+x_{2}}{2},\ \ \ \ \ \ y=\frac{y_{1}+y_{2}}{2}$

Using above formula, we get

Area of first triangle = 4 unit

and, the midpoints of side of first triangle is (1, 0) , (1, 0) and (1, 2)

which is not a triangle, it is straight line.

So, ratio can't be find.

Answer 2

Given that,

Point of triangle is

A = (0,-1)

B = (2,1)

C = (0,3)

We need to calculate the area of triangle ABC

Using formula of area

$A_{ABC}=\dfrac{1}{2}(x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2}))$

Put the value into the formula

$A_{ABC}=\dfrac{1}{2}(0(1-3)+2(3+1)+0(-1-1))$

$A_{ABC}=4\ square\ units$

We need to calculate the coordinates of midpoint of AB, AC and BC

Using formula of coordinate

Coordinate of E $(x,y)=\dfrac{0+2}{2},\dfrac{-1+1}{2}$

Coordinate of E $(x,y)=(1,0)$

Coordinate of F $(x,y)=\dfrac{2+0}{2},\dfrac{1+3}{2}$

Coordinate of F $(x,y)=(1,2)$

Coordinate of G $(x,y)=\dfrac{0+0}{2},\dfrac{-1+3}{2}$

Coordinate of G $(x,y)=(0,1)$

We need to calculate the area of  new triangle GEF

Using formula of area

$A_{GEF}=\dfrac{1}{2}(x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2}))$

Put the value into the formula

$A_{GEF}=\dfrac{1}{2}(1(2-1)+1(1-0)+0(0-2))$

$A_{GEF}=1\ square\ units$

We need to calculate the ratio of triangle formed by joining the points

Using area of both triangles

$\dfrac{A_{GEF}}{A_{ABC}}=\dfrac{1}{4}$

Hence, The ratio of triangle formed by joining the points is 1:4

The area of triangle formed by joining the mid points of sides of triangle is 1 square unit