SME
Grade 10 Math · Problem type
Prove a parallelogram with slopes
Use this for “Show that A(−3, −1), B(−1, 3), C(5, 4), D(3, 0) is a parallelogram.”
Chart
Slopes: AB 2, BC 1/6, CD 2, DA 1/6.
AB ∥ CD: equal slopes, so parallel.
BC ∥ DA: equal slopes, so parallel.
This is an example. Type your own values to start a new problem.
xA
x of A
example
yA
y of A
example
xB
x of B
example
yB
y of B
example
xC
x of C
example
yC
y of C
example
xD
x of D
example
yD
y of D
example
mAB
Slope of AB
worked out (not typed)
mBC
Slope of BC
worked out (not typed)
mDC
Slope of DC
worked out (not typed)
mAD
Slope of AD
worked out (not typed)
v
Whether ABCD is a parallelogram
worked out (not typed)
Formulas
mAB
=
yB − yA
xB − xA
2
=
3 − (−1)
−1 − (−3)
mBC
=
yC − yB
xC − xB
1
6
=
4 − 3
5 − (−1)
mDC
=
yC − yD
xC − xD
2
=
4 − 0
5 − 3
mAD
=
yD − yA
xD − xA
1
6
=
0 − (−1)
3 − (−3)
v
:
compare
mAB
with
mDC
and
mBC
with
mAD
2
=
2
and
1
6
=
1
6
,
so
yes
Assumptions
•
Sides with equal slopes are parallel.
•
Both pairs of opposite sides parallel: the quadrilateral is a parallelogram.
•
A vertical side has no slope; two vertical sides are still parallel.
Step-by-step
WE KNOW
xA = −3, yA = −1, xB = −1, yB = 3, xC = 5, yC = 4, xD = 3, yD = 0
FIND
slope of AB (mAB), slope of BC (mBC), slope of DC (mDC), slope of AD (mAD), whether ABCD is a parallelogram (v)
Step 1 · Find slope of AB (mAB)
Use mAB = (yB − yA) ÷ (xB − xA)
Rise over run from A to B.
mAB
=
3 − (−1)
−1 − (−3)
mAB
=
4
2
mAB = 2
Step 2 · Find slope of BC (mBC)
Use mBC = (yC − yB) ÷ (xC − xB)
Rise over run from B to C.
mBC
=
4 − 3
5 − (−1)
mBC
=
1
6
mBC
=
1
6
Step 3 · Find slope of DC (mDC)
Use mDC = (yC − yD) ÷ (xC − xD)
Rise over run from D to C.
mDC
=
4 − 0
5 − 3
mDC
=
4
2
mDC = 2
Step 4 · Find slope of AD (mAD)
Use mAD = (yD − yA) ÷ (xD − xA)
Rise over run from A to D.
mAD
=
0 − (−1)
3 − (−3)
mAD
=
1
6
mAD
=
1
6
Step 5 · Find whether ABCD is a parallelogram (v)
Use v: compare mAB with mDC and mBC with mAD
mAB = mDC and mBC = mAD: both pairs of opposite sides are parallel, so ABCD is a parallelogram.
2
=
2
and
1
6
=
1
6
:
yes
v
=
compare
2
with
2
and
1
6
with
1
6
v
=
yes
Check
Put the numbers back in. Both sides should match.
2
=
3 − (−1)
−1 − (−3)
✓
1
6
=
4 − 3
5 − (−1)
✓
2
=
4 − 0
5 − 3
✓
1
6
=
0 − (−1)
3 − (−3)
✓
2
=
2
and
1
6
=
1
6
,
so
yes
✓
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