A quadrilateral has vertices A(-1, -2), B(1, 2), C(9, -2), and D(7, -6). Without graphing, determine whether this quadrilateral is a rectangle.

Question

Note: vertices are listed in clockwise order

Note: a quadrilateral is a 4-sided shape

Note: a rectangle has two sets of parallel lines

(1 mark for finding slope of AB, 1 mark for finding slope of BC, 1 mark for finding slope of CD, 1 mark for finding slope of AD, 2 marks for stating if the quadrilateral is a rectangle and how you know).

Answer 1

If the four vertices form a rectangle, then there are two sets of parallel lines and those sets are perpendicular to each other. Your quadrilateral (four-sided shape) has sides AB, BC, CD, and DA. For the quadrilateral to be a rectangle, then AB and CD are parallel and, thus, have the same slope. BC and DA are parallel and, thus, have the same slope. One more thing, the slope of the first pair is the negative reciprocal of the slope of the second pair, meaning they are perpendicular, meeting at right angles and forming a rectangle.

y_A – y_B –2 – 2 –4

Slope AB = _______ = _______ = ______ = 2

x_A – x_B –1 – 1 –2

y_B – y_C 2 — (–2) 2 + 2 4 –1

Slope BC = _______ = __________ = _______ = ______ = _____

x_B – x_C 1 – 9 –8 –8 2

y_C – y_D

Slope CD = _________

x_C – x_D

y_D – y_A

Slope DA = _______

x_D – x_A

I will let you calculate the last two slopes and check to see if we have pairs of parallel line segments and if those pairs are perpendicular to each other. That's YOUR homework!!

Answer 2

AB:

Slope, m_ab = changey/changex = (2-(-2))/(1-(-1) = 4/2 = 2

BC:

m_bc = (-2-2)/((9-1) = -4/8 = -1/2

CD:

m_cd = (-6 -(-2)/(7-9) = -4/-2 = 2

AD:

m_ad = (-6-(-2))/(7-(-1)) = -4/8 = -1/2

m_ab = m_cd = 2

m_bc = m_ad = -1/2

ABCD is a rectangle because AB is parallel with CD and BC is parallel with AD

Answer 3

It looks like you will be given marks based on finding the slopes on each side of the rectangle. Since rectangles have right angles on each corner, we need to check if the slopes are perpendicular to each other. We also need to make sure we have two pairs of parallel lines.

The first step is to find the slopes of each side using the slope formula: (y₂-y₁)/(x₂-x₁

For AB, use points A(-1, -2), B(1, 2)

Slope of AB: (2-(-2))/(1-(-1))

=(4)/(2)

= 2

For BC, use points B(1, 2), C(9, -2)

Slope of BC: (-2-2)/(9-1)

=(-4)/(8)

= -1/2

For CD, use points C(9, -2), D(7, -6)

Slope of CD: (-6-(-2))/(7-9)

= (-4)/(-2)

= 2

For AD, use points A(-1, -2), D(7, -6)

Slope of AD: (-6-(-2))/(7-(-1))

= -4/8

= -1/2

Based on our slopes, you can see that we have 2 pairs of slopes that are the same, which means they are parallel. You also have slopes that are both opposite and reciprocal to another slope, which makes them perpendicular. If they are perpendicular that means they form right angles.

So since you have two pairs of parallel sides and right angles at all intersections, this is a rectangle.