Question

The coordinates of the vertices of triangle RST are R(-3,1), S(-1,4), and T(3,1). Which statement correctly describes whether triangle RST is a right triangle?

A) Triangle △RST is a right triangle because the slopes of the sides RS and ST are negative reciprocals.
B) Triangle △RST is not a right triangle because it does not have a side with a length of 5 units.
C) Triangle △RST is a right triangle because the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side.
D) Triangle △RST is not a right triangle because the angles at vertices R, S, and T do not add up to 180 degrees.

Answer 1

Triangle RST is a right triangle because the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side.

Step-by-step explanation:

The statement that correctly describes whether triangle RST is a right triangle is option C) Triangle △RST is a right triangle because the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side.

To determine if a triangle is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the two shorter sides.

In triangle RST, the lengths of the sides are RS = sqrt((4-1)^2 + (-1+3)^2) = sqrt(3^2 + 2^2) = sqrt(9 + 4) = sqrt(13), ST = sqrt((3-(-1))^2 + (1-4)^2) = sqrt(4^2 + (-3)^2) = sqrt(16 + 9) = sqrt(25) = 5, and TR = sqrt((-3-3)^2 + (1-1)^2) = sqrt((-6)^2 + 0^2) = sqrt(36) = 6.

Applying the Pythagorean theorem, we have that RS^2 + ST^2 = TR^2, or (sqrt(13))^2 + 5^2 = 6^2. Simplifying, 13 + 25 = 36, which is true. Therefore, triangle RST satisfies the condition for a right triangle.