# Thread: Triangles and graphs

1. ## Triangles and graphs

a) prove that the triangle with the vertices at the points:

A (2,1)
B(4,5)
C(0,7)

is right angled

b) calculate the area of this triangle

thanks for any help
x

2. Originally Posted by pop_91
a) prove that the triangle with the vertices at the points:

A (2,1)
B(4,5)
C(0,7)

is right angled

b) calculate the area of this triangle

thanks for any help
x
to a)

Calculate the slope of AB and BC:

$\displaystyle m_{AB}=\dfrac{5-1}{4-2}= 2$

$\displaystyle m_{BC}=\dfrac{5-7}{4-0}=-\frac12$

Since $\displaystyle m_{AB} \cdot m_{BC} = -1 ~\implies~AB \perp BC$

to b)
A right triangle is a half rectangle. Therefore the area of the triangle is:

$\displaystyle a_{ABC} = \frac12 \cdot AB \cdot BC$

Use the distance formula to calculate the length of the sides:

$\displaystyle AB=\sqrt{(4-2)^2+(5-1)^2}=2\sqrt{5}$

$\displaystyle BC=\sqrt{(4-0)^2+(5-7)^2}=2\sqrt{5}$

Therefore

$\displaystyle a_{ABC} = \frac12 \cdot 2\sqrt{5} \cdot 2\sqrt{5} = 10$