Question

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Given the vertices A(2,-2) B(0,4) and C(-4,-4): Classify the triangle by its sides: scalene, isosceles, or equilateral (Is the triangle scalene, isosceles, or equilateral?) Classify the triangle by its angles: acute, obtuse, or right (Is the triangle acute, obtuse, or right?)

Answer 1

Answer:

• sides: isosceles
• angles: right

Step-by-step explanation:

At first, lets find the distance from one point to the other.

$d(A,B)=\sqrt{(x_{A}-x_{B} )^2 +(y_{A}-y_{B} )^2 } \\d(A,B)=\sqrt{(2-0 )^2 +(-2-4 )^2 } \\d(A,B)=\sqrt{4 +36 } = \sqrt{40 } = \sqrt{4 * 10 } = 2\sqrt{10}$

The same way, you can find that:

$d(A,C)= 2\sqrt{10}\\d(A,B)= 2\sqrt{20}$

Since the triangle has only its two sides equal to each other (AB = AC),

it's an isosceles one.

As long as the angles are concerned, we gotta "cheat". In other words, we see that:

$AB^2 + AC^2 = (\sqrt{40})^2 + (\sqrt{40})^2 = 40 + 40 = 80\\ AB^2 + AC^2 = 80 = (2\sqrt{20})^2 = BC^2\\AB^2 + AC^2 = BC^2$

By the inverse of pythagorean theorem, we know that if the above equation holds, then we have a right-angled triangle. That is to say that,

A = 90

Since its an isosceles triangle, AB = AC,

B = C = (180 - A)/2 = 45 degrees