Question

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

A) ARST is a right triangle because RS is perpendicular to RT.
B) ARST is not a right triangle because the sum of the squares of its sides does not equal the square of the hypotenuse.
C) ARST is a right triangle because the slope of RS is equal to the negative reciprocal of the slope of RT.
D) ARST is not a right triangle because the angles at R, S, and T are not 90 degrees.

Answer 1

ARST is not a right triangle because the sum of the squares of its sides does not equal the square of the hypotenuse.

Step-by-step explanation:

To determine if ARST is a right triangle, we need to check if the lengths of its sides satisfy the Pythagorean theorem. The Pythagorean theorem states that for a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the hypotenuse. In this case, we can calculate the lengths of the sides and check if they satisfy this condition.

Using the coordinates given, we can calculate the lengths of the sides:

• Length of RS = √((-1 - (-3))^2 + (4 - 1)^2) = √(2^2 + 3^2) = √(4 + 9) = √13
• Length of RT = √((3 - (-3))^2 + (1 - 1)^2) = √(6^2 + 0^2) = 6
• Length of ST = √((-1 - 3)^2 + (4 - 1)^2) = √((-4)^2 + 3^2) = √(16 + 9) = √25 = 5

Now, let's check if the lengths of the sides satisfy the Pythagorean theorem:

√RS^2 + √ST^2 = √13^2 + 5^2 = 13 + 25 = 38

√RT^2 = √6^2 = 6

Since the sum of the squares of RS and ST (38) is not equal to the square of RT (6), we can conclude that ARST is not a right triangle. Therefore, option B) ARST is not a right triangle because the sum of the squares of its sides does not equal the square of the hypotenuse is the correct statement.