Triangle has coordinates $(-4, 3), (-4, 5)$ , and $(1,5)$ . How do you prove that the triangle is right-angled?

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Answer 1 · 2017-06-02T10:06:13

See the proof below

The points are

$A=(-4,3)$

$B=(-4,5)$

$C=(1,5)$

The lengths of

$AB=2$

$BC=5$

Firstly, we calculate $AC$ , assuming that the angle $hat(ABC)$ is a right angle

$AC^2=AB^2+BC^2$

$AC^2=4+25=29$

$AC=sqrt29$

Secondly, we calculate $AC$ from the coordinates of points $A=(-4,3)$ and $C=(1,5)$

$AC=sqrt((-4-1)^2+(3-5)^2)$

$=sqrt(5^2+2^2)$

$=sqrt29$

As the distance, $AB$ is the same by $2$ methods of calculation.

We conclude, that as

$AC^2=AB^2+BC^2$

the triangle $ABC$ , is right angle at $hatB$

$QED$

Triangle has coordinates (-4, 3), (-4, 5)(-4, 3), (-4, 5), and (1,5)(1,5). How do you prove that the triangle is right-angled?