Question

In the coordinate plane, the vertices of triangle PAT are P(-1, -6), A(-4, 5), and T(5, -2). Show that triangle PAT is isosceles but not equilateral

Answer 1

Answer: To determine if a triangle is isosceles, we need to check if at least two sides are equal in length. In the coordinate plane, we can find the length of a side by using the distance formula:

distance between two points (x1, y1) and (x2, y2) = √((x2 - x1)^2 + (y2 - y1)^2)

Applying this formula to the three sides of triangle PAT, we have:

PA = √((-4 - (-1))^2 + (5 - (-6))^2) = √((-3)^2 + 11^2) = √(9 + 121) = √(130)

PT = √((5 - (-1))^2 + (-2 - (-6))^2) = √((6)^2 + 4^2) = √(36 + 16) = √(52)

AT = √((5 - (-4))^2 + (-2 - 5)^2) = √((9)^2 + 7^2) = √(81 + 49) = √(130)

We see that PA and AT are equal in length, so triangle PAT is isosceles.

To determine if a triangle is equilateral, we need to check if all sides are equal in length. Since PA and AT are equal in length but PT is not equal to either PA or AT, we can conclude that triangle PAT is isosceles but not equilateral.

Explanation: