Question

Use the distance formula and the Pythagorean Theorem to decide if the triangle determined by the points A(-5,-1),B(2,1), and C(-1,6) is a right triangle.

Answer 1

Using the distance formula and the Pythagorean Theorem, we can determine that the triangle determined by the points A(-5,-1), B(2,1), and C(-1,6) is a right triangle.

Step-by-step explanation:

To determine if the triangle determined by the points A(-5,-1), B(2,1), and C(-1,6) is a right triangle, we can use the distance formula and the Pythagorean Theorem.

The distance between points A and B is calculated as follows:

dAB = sqrt((2 - (-5))^2 + (1 - (-1))^2) = sqrt(49 + 4) = sqrt(53).

The distance between points A and C is calculated as follows:

dAC = sqrt((-1 - (-5))^2 + (6 - (-1))^2) = sqrt(16 + 49) = sqrt(65).

The distance between points B and C is calculated as follows:

dBC = sqrt((-1 - 2))^2 + (6 - 1))^2) = sqrt(9 + 25) = sqrt(34).

Now, we will check if the sum of the squares of the two shorter sides equals the square of the longest side:

sqrt(53)^2 + sqrt(34)^2 = sqrt(65)^2.

Since both sides of the equation are equal, we can conclude that the triangle determined by the points A(-5,-1), B(2,1), and C(-1,6) is a right triangle.