Question #350448

Show that the points (-2,0), (2,3) and (5,-1) are the vertices of a right triangle. And find its area

1
Expert's answer
2022-06-14T11:22:49-0400

A(2,0),B(2,3),C(5,1)A(-2,0), B(2,3), C(5, -1)

BA=22,03=4,3\overrightarrow{BA}=\langle-2-2, 0-3\rangle=\langle-4, -3\rangle

BC=52,13=3,4\overrightarrow{BC}=\langle5-2, -1-3\rangle=\langle3, -4\rangle

BABC=4(3)+(3)(4)=0\overrightarrow{BA}\cdot\overrightarrow{BC}=-4(3)+(-3)(-4)=0

Then BABC,\overrightarrow{BA}\perp\overrightarrow{BC}, and we see that the triangle ABCABC with vertices A(2,0),B(2,3),C(5,1)A(-2,0), B(2,3), C(5, -1) is a right triangle.


BA=(4)2+(3)2=5|\overrightarrow{BA}|=\sqrt{(-4)^2+(-3)^2}=5

BC=(3)2+(4)2=5|\overrightarrow{BC}|=\sqrt{(3)^2+(-4)^2}=5

AreaABC=12BABAArea_{ABC}=\dfrac{1}{2}|\overrightarrow{BA}||\overrightarrow{BA}|

=12(5)(5)=12.5(units2)=\dfrac{1}{2}(5)(5)=12.5({units}^2)


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