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