2) Prove that PAL with coordinates P(0-5),A(3,-5), L(3,-2) is an isosceles right triangle.

An isosceles triangle has two sides which are the same length.  With coordinates it's often easier and just as good to verify the same squared length; we skip the square root.  But here the two sides we're interested in are parallel to the axes so the length calculation is easy.

P(0,-5), A(3,-5), L(3,-2)

|PA| = 3, the difference in the x coordinates, because the y coordinate are the same

|AL| = |-2 - -5| = 3, the difference in the y coordinates, because the x coordinates are the same

|PA| = |AL| so we have an isoscles triangle.

PA is parallel to the x axis and AL is parallel to the y axis PAL has a right angle at A.  

So we have an isosceles right triangle, one of the two tired triangle of trig.

If that's not enough of a proof, we can verify we have a right triangle a few different ways.  Let's just do it by the Pythagorean Theorem.

A is a right angle iff |PA|² + |AL|² = |PL|²

The left side is |PA|² + |AL|² = 3² + 3² = 18

|PL|² = (3 - 0)² + (-2 - -5)² = 18

They're the same so we have a right angle.  √

the correct answer is d

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