c To tell if the quadrilateral is a rectangle, we need to find if the angles are right angles. Because the opposite sides have equal lengths, if one angle is

9 0^{\circ},

all of them are. Let's see if angle

E

is right by checking if the sides are perpendicular. For this, we need to know the slopes of the sides

\overline{E F}

and

\overline{E H} .

Let's find the slope of

\overline{E F} .

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(- 7, 3)$ & $(- 3, 6)$

m_{\overline{E F}} = \frac{3 - 6}{- 7 - ( - 3 )}

▼

Simplify right-hand side

SubNeg

$a - (- b) = a + b$

m_{\overline{E F}} = \frac{3 - 6}{- 7 + 3}

AddSubTerms

Add and subtract terms

m_{\overline{E F}} = \frac{- 3}{- 4}

DivNegNeg

$\frac{- a}{- b} = \frac{a}{b}$

m_{\overline{E F}} = \frac{3}{4}

Now, let's calculate the slope of

\overline{E H} .

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(3, - 2)$ & $(- 3, 6)$

m_{\overline{E H}} = \frac{- 2 - 6}{3 - ( - 3 )}

▼

Simplify right-hand side

SubNeg

$a - (- b) = a + b$

m_{\overline{E H}} = \frac{- 2 - 6}{3 + 3}

AddSubTerms

Add and subtract terms

m_{\overline{E H}} = \frac{- 8}{6}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

m_{\overline{E H}} = \frac{- 4}{3}

Two lines are perpendicular if their slopes are negative reciprocals. This is true if the product of the slopes is

- 1 .

m_{\overline{E F}} \cdot m_{\overline{E H}} = \frac{3}{4} \cdot \frac{- 4}{3} = - 1

Therefore, these two sides of the quadrilateral are perpendicular and, by the reasoning explained above, so are the remaining sides.

Exercise