Question

quadrilateral (abcd) has the following vertices:

• (a(-6, -2))
• (b(-4, 4))
• (c(8, 1))
• (d(6, -6))

also, (angle a) is a right angle.
is quadrilateral (abcd) a rectangle, and why?
choose 1 answer:
a yes, because (ab = bc = cd = ad), and (angle a) is a right angle.
b yes, because opposite sides are parallel, and (angle a) is a right angle.
c no, because (ab) is not parallel to (cd).
d no, because (ab) is shorter than (bc).

Explanation:

To determine if quadrilateral \(ABCD\) is a rectangle, we need to check the properties of a rectangle: opposite sides are parallel and all angles are right angles (or, since we know \(\angle A\) is a right angle, we can check if opposite sides are parallel).

Step 1: Recall the slope formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 2: Calculate the slope of \(AB\)

For points \(A(-6, -2)\) and \(B(-4, 4)\):

$$ m_{AB} = \frac{4 - (-2)}{-4 - (-6)} = \frac{6}{2} = 3 $$

Step 3: Calculate the slope of \(BC\)

For points \(B(-4, 4)\) and \(C(8, 1)\):

$$ m_{BC} = \frac{1 - 4}{8 - (-4)} = \frac{-3}{12} = -\frac{1}{4} $$

Step 4: Calculate the slope of \(CD\)

For points \(C(8, 1)\) and \(D(6, -6)\):

$$ m_{CD} = \frac{-6 - 1}{6 - 8} = \frac{-7}{-2} = \frac{7}{2} $$

Step 5: Calculate the slope of \(AD\)

For points \(A(-6, -2)\) and \(D(6, -6)\):

$$ m_{AD} = \frac{-6 - (-2)}{6 - (-6)} = \frac{-4}{12} = -\frac{1}{3} $$

Wait, but we know \(\angle A\) is a right angle, so \(AB\) and \(AD\) should be perpendicular (their slopes should multiply to \(-1\)). Let's check: \(m_{AB} \times m_{AD} = 3 \times (-\frac{1}{3}) = -1\), so they are perpendicular (good, \(\angle A\) is right). Now, for a rectangle, \(AB \parallel CD\) and \(BC \parallel AD\). Let's check \(AB\) and \(CD\) slopes: \(m_{AB} = 3\), \(m_{CD} = \frac{7}{2}\). Since \(3
eq \frac{7}{2}\), \(AB\) is not parallel to \(CD\). Wait, but maybe I made a mistake? Wait, no, let's re - check the slope of \(CD\). Wait, \(C(8,1)\) and \(D(6, - 6)\): \(y_2 - y_1=-6 - 1=-7\), \(x_2 - x_1 = 6 - 8=-2\), so slope is \(\frac{-7}{-2}=\frac{7}{2}\), correct. Slope of \(AB\) is 3. So \(AB\) and \(CD\) have different slopes, so they are not parallel.

Now let's analyze the options:

• Option A: Says \(AB = BC=CD = AD\). Let's calculate lengths. Length of \(AB\): \(\sqrt{(-4 + 6)^2+(4 + 2)^2}=\sqrt{4 + 36}=\sqrt{40}\). Length of \(BC\): \(\sqrt{(8 + 4)^2+(1 - 4)^2}=\sqrt{144 + 9}=\sqrt{153}\). So \(AB

eq BC\), so A is wrong.

• Option B: Says opposite sides are parallel, but we saw \(AB\) is not parallel to \(CD\), so B is wrong.

• Option C: Says No, because \(AB\) is not parallel to \(CD\). This matches our calculation.

• Option D: Says No, because \(AB\) is shorter than \(BC\). But the key property of a rectangle is parallel opposite sides, not side lengths comparison in this way, so D is wrong.

Answer:

C. No, because \(AB\) is not parallel to \(CD\).