Question

the vertices of a triangle are (2,1),(5,2) and (4,4). find the length of the perpendicular from the first vertex to the opposite side​

Answer 1

Answer:

3.13 units

Step-by-step explanation:

A=$(x_1,y_1)=(2,1)$

B=$(x_2,y_2)=(5,2)$

C=$(x_3,y_3)=(4,4)$

First find length of sides of triangle

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

A=$(x_1,y_1)=(2,1)$

B=$(x_2,y_2)=(5,2)$

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

$AB=3.162$

B=$(x_1,y_1)=(5,2)$

C=$(x_2,y_2)=(4,4)$

$BC=\sqrt{(4-5)^2+(4-2)^2}$

$BC=2.236$

A=$(x_1,y_1)=(2,1)$

C=$(x_2,y_2)=(4,4)$

$AC=\sqrt{(4-2)^2+(4-1)^2}$

$AC=3.605$

Now to find area of triangle

a = 3.162

b = 2.236

c =3.605

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

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

$s=\frac{3.162+2.236+3.605}{2}$

$s=4.5015$

$Area=\sqrt{4.5015(4.5015-3.162)(4.5015-2.236)(4.5015-3.605)}$

$Area=3.4995$

The base corresponding to point A is BC

So, To find length  of the perpendicular from the first vertex to the opposite side​

Area of triangle = $\frac{1}{2} \times Base \times Height$

$3.4995=\frac{1}{2} \times 2.236 \times Height$

$Height=3.13$

Hence the length of the perpendicular from the first vertex to the opposite side​ is 3.13 units