Question

find the area of triangle formed by points (8,-5),(-2,-7),(5,1) by using herons formula

Answer 1

herons formula is
area = sq root ( s(s-a) (s- b) (s- c ) )
where a b c are side length find using distance formula
sq root ( (x1 - x2)^2 + (y1-y2)^2 )
s=(a + b+c)/2

Answer 2

Answer:

32.947 sq. units.

Step-by-step explanation:

A=(8,-5)

B = (-2,-7)

C = (5,1)

Now find the sides of triangle AB,BC,AC

To find AC use distance formula :

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$(x_1,y_1)=(8,-5)$

$(x_2,y_2)=(-2,-7)$

Substitute the values in the formula :

$AB=\sqrt{(-2-8)^2+(-7+5)^2}$

$AB=\sqrt{(-10)^2+(-2)^2}$

$AB=\sqrt{100+4}$

$AB=\sqrt{104}$

$AB=10.19$

To Find BC

$(x_1,y_1)=(-2,-7)$

$(x_2,y_2)=(5,1)$

Substitute the values in the formula :

$BC=\sqrt{(5+2)^2+(1+7)^2}$

$BC=\sqrt{(7)^2+(8)^2}$

$BC=\sqrt{49+64}$

$BC=\sqrt{113}$

$BC=10.63$

To Find AC

$(x_1,y_1)=(8,-5)$

$(x_2,y_2)=(5,1)$

Substitute the values in the formula :

$AC=\sqrt{(5-8)^2+(1+5)^2}$

$AC=\sqrt{(-3)^2+(6)^2}$

$AC=\sqrt{9+36}$

$AC=\sqrt{45}$

$AC=6.70$

So, sides of triangle :

a =10.19

b=10.63

c= 6.70

Now to find area:

$Area = \sqrt{s(s-a)(s-b)(s-c)}$

Where $s = \frac{a+b+c}{2}$

a,b,c are the side lengths of triangle  

Now substitute the values :

$s = \frac{10.19+10.63+6.70}{2}$

$s =13.76$

$Area = \sqrt{13.76(13.76-10.19)(13.76-10.63)(13.76-6.70)}$

$Area = 32.947$

Hence the area of the given triangle is 32.947 sq. units.