We are given the coordinates of the vertices of a quadrilateral MATH: $M(-1, 7)$, $A(3, 5)$, $T(2, -7)$, and $H(-6, -3)$. We need to prove that MATH is a trapezoid. A trapezoid is a quadrilateral with at least one pair of parallel sides. To prove that MATH is a trapezoid, we need to show that one pair of opposite sides has the same slope (i.e., are parallel).

1. Problem Description

We are given the coordinates of the vertices of a quadrilateral MATH: $M(-1, 7)$, $A(3, 5)$, $T(2, -7)$, and $H(-6, -3)$. We need to prove that MATH is a trapezoid. A trapezoid is a quadrilateral with at least one pair of parallel sides. To prove that MATH is a trapezoid, we need to show that one pair of opposite sides has the same slope (i.e., are parallel).

2. Solution Steps

First, we need to calculate the slopes of the four sides of the quadrilateral. The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Slope of MA:

$m_{MA} = \frac{5 - 7}{3 - (-1)} = \frac{-2}{4} = -\frac{1}{2}$

Slope of AT:

$m_{AT} = \frac{-7 - 5}{2 - 3} = \frac{-12}{-1} = 12$

Slope of TH:

$m_{TH} = \frac{-3 - (-7)}{-6 - 2} = \frac{4}{-8} = -\frac{1}{2}$

Slope of HM:

$m_{HM} = \frac{7 - (-3)}{-1 - (-6)} = \frac{10}{5} = 2$

Since $m_{MA} = m_{TH} = -\frac{1}{2}$, the sides MA and TH are parallel. Since only one pair of sides needs to be parallel for the quadrilateral to be a trapezoid, we have shown that quadrilateral MATH is a trapezoid.

3. Final Answer

Quadrilateral MATH is a trapezoid because the sides MA and TH are parallel, as they have the same slope of $-\frac{1}{2}$.