Question

43. Prove that, the points (2, -2), (-2, 1) and (5, 2) are the vertices of a right triangle. Also find the area of trangle​

Answer 1

Step-by-step explanation:

Given:-

the points (2, -2), (-2, 1) and (5, 2)

To find:-

Prove that, the points (2, -2), (-2, 1) and (5, 2) are the vertices of a right triangle. Also find the area of trangle.

Solution:-

Given points are (2, -2), (-2, 1) and (5, 2)

To prove that the vertices A,B,C of a right triangle then we have to prove that BC²=AB²+AC² i.e. The square of the Hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras theorem)

I)Distance between A and B:-

Let (x1,y1)=(2,-2)=>x1=2 and y1=-2

Let (x2, y2)=(-2,1)=>x2=-2 and y2=1

Distance between the two points (x1,y1) and (x2,y2) is √{(x2-x1)²+(y2-y2)²} units

=>√{(-2-2)²+(1+2)²}

=>√{(-4)²+(3)²}

=>√(16+9)

=>√25

=>5 units

Distance between A and B=AB=5 units

AB²=25 units -----------(1)

ii)Distance between B and C :-

Let (x1, y1)=(-2,1)=>x1=-2 and y1=1

Let (x2, y2)=(5,2)=>x2=5 and y2=2

Distance between the two points (x1,y1) and (x2,y2) is √{(x2-x1)²+(y2-y2)²} units

=>√{(5+2)²+(2-1)²}

=>√{(7)²+(1)²}

=>√(49+1)

=>√50 units

Distance between B and C =BC=√50 units

BC²=50 units ----------(2)

iii) Distance between C and A:-

Let (x1,y1)=(5,2)=>x1=5 and y1=2

Let (x2,y2)=(2,-2)=>x2=2 and y2=-2

Distance between the two points (x1,y1) and (x2,y2) is √{(x2-x1)²+(y2-y2)²} units

=>√{(2-5)²+(-2-2)²}

=>√{(-3)²+(-4)²}

=>√{9+16}

=>√25

=>5 units

Distance between C and A = AC=5 units

AC²=25 units----------(3)

From (1),(2)&(3)

50=25+25

BC²=AB²+AC²

Here, BC is the hypotenuse and AB and AC are the other two sides.

Area of the triangle:-

Given triangle is a right angled triangle

Area of a right angled triangle =ab/2 sq.units

Here, a=AB=5 units ,b=AC=5 units

Area =5×5/2

=>Area =25/2

=>12.5 sq.units

Answer:-

1)the points (2, -2), (-2, 1) and (5, 2) are the vertices of a right triangle.

2)Area of the right angled triangle=12.5 sq.units

Used formulae:-

• Pythagoras theorem:-

"The square of the Hypotenuse is equal to the sum of the squares of the other two sides".

• Distance formula:-

Distance between the two points (x1,y1) and (x2,y2) is √{(x2-x1)²+(y2-y2)²} units

• Area of a right angled triangle:-

Area of a right angled triangle =ab/2 sq.units

where a and b are the two sides other than hypotenuse.