Questions: Determine if the given points form the vertices of a right triangle. M(3,-3), P(5,-5), and Q(-1,-7) The given points do not form the vertices of a right triangle. The given points form the vertices of a right triangle.

Solution Steps

To determine if the given points form the vertices of a right triangle, we can use the distance formula to calculate the lengths of the sides of the triangle formed by these points. Then, we check if the Pythagorean theorem holds for any combination of these sides. If it does, the points form a right triangle.

To determine if the points \( M(3, -3) \), \( P(5, -5) \), and \( Q(-1, -7) \) form a right triangle, we first calculate the distances between each pair of points using the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• Distance \( MP \): \[ MP = \sqrt{(5 - 3)^2 + (-5 + 3)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2.828 \]

• Distance \( PQ \): \[ PQ = \sqrt{(-1 - 5)^2 + (-7 + 5)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{40} = 6.325 \]

• Distance \( QM \): \[ QM = \sqrt{(3 + 1)^2 + (-3 + 7)^2} = \sqrt{4^2 + 4^2} = \sqrt{32} = 5.657 \]

To verify if these points form a right triangle, we check if the Pythagorean theorem holds for any combination of these side lengths. The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the following must be true:

\[ a^2 + b^2 = c^2 \]

• Check if \( MP^2 + QM^2 = PQ^2 \): \[ 2.828^2 + 5.657^2 = 6.325^2 \] \[ 7.999 + 31.999 = 40.000 \]

Since the equation holds, the points form a right triangle.

Final Answer