Download Coordinate geometry: working with slopes

Honors Math 2
Name: ____________________
Coordinate geometry: working with slopes
This assignment begins our unit on coordinate geometry. The unit objective is to use coordinate
calculations to solve geometric problems. Today’s problem set focuses on using slopes to
reach conclusions about lines being parallel or perpendicular. Your homework for is to finish this
problem set.
Here is a reminder of what it means for two lines to be parallel and for two lines to be
perpendicular, from algebraic and geometric perspectives.
algebraic meaning
Two lines are parallel if their slopes are equal.
Two lines are perpendicular if the product of
their slopes is –1. (In other words, their slopes
are opposite reciprocals of each other.)
geometric meaning
Two lines are parallel if they do not intersect.
Two lines are perpendicular if they form a
right angle at their intersection.
Problems
In the following problem set, you will use slope calculations to analyze the special properties of
shapes. For example, a quadrilateral (four-sided shape) might be a rectangle, a parallelogram,
a trapezoid, or none of these, depending on the slopes of its sides.
1. Consider triangle ABC with vertices A(6, 5), B(2, –1), and C(–10, 7).
a. Draw the triangle on graph paper.
b. Each side of the triangle is part of a line. Write an equation for each line.
c. What is special about triangle ABC? Justify your answer using slopes.
2. You can calculate the midpoint of a line segment by thinking of the two endpoints as a
data set and finding their average. The midpoint’s x-coordinate is the average of the
x-coordinates of the endpoints, and the midpoint’s y-coordinate is the average of the
y-coordinates of the endpoints.
a. Find the midpoint of the line segment from (–7, 5) to (3, 8).
b. Generalize: Find the midpoint of the line segment from (x1, y1) to (x2, y2).
3. Again use triangle ABC from problem 1, where A(6, 5), B(2, –1), and C(–10, 7).
a. Let D be the midpoint of BC, let E be the midpoint of AC, and let F be the midpoint
of AB. Find the coordinates of D, of E, and of F.
b. Add points D, E, and F to your graph from problem 1. Use the graph to check that your
answers to part a are correct.
c. Think of points D,E, and F as forming a new triangle, the “midpoint triangle.”
Calculate the slopes of the three sides of the triangle DEF.
d. How do the slopes for triangle ABC compare to the slopes for triangle DEF?
e. Generalize: In general, how do the slopes of the midpoint triangle appear to be related to
the slopes of the original triangle?
f. Take it further: Prove that your generalization holds true for every triangle. Hint: Use
the coordinates A(x1, y1), B(x2, y2), C(x3, y3) and perform the same calculations again.
page 2
REMINDERS:
A parallelogram is a quadrilateral (four-sided shape) with two pairs of parallel sides.
A trapezoid is a quadrilateral with only one pair of parallel sides.
A rectangle is a quadrilateral with right angles at all four vertices (corners).
4. Consider quadrilateral WXYZ with vertices W(–2, 3), X(–1, 8), Y(9, 6), and Z(8, 1).
a. Is this quadrilateral a parallelogram? Justify your answer using slopes.
b. Is this quadrilateral a trapezoid? Justify your answer using slopes.
c. Is this quadrilateral a rectangle? Justify your answer using slopes.
5. Consider quadrilateral STUV with vertices S(0, 2), T(2, 6), U(6, 4), and V(2, 1).
a. How many right angles does STUV have? Justify your answer using slopes.
b. Prove that STUV is a trapezoid.
6. Consider quadrilateral OPQR with O(2, 0), P(8, 4), Q(5, 8), and R(2, 6).
a. Using slopes, determine whether quadrilateral OPQR is a parallelogram, a trapezoid,
a rectangle, or none of these.
b. Let K, L, M, and N stand for the midpoints of OP, PQ, QR, and RO.
Calculate the coordinates of these four points.
c. Graph the quadrilateral and its midpoints on graph paper. Use your graph to check your
answers to part b.
d. Think of points K, L, M, and N as forming a new quadrilateral, the “midpoint
quadrilateral.” Using slopes, determine whether quadrilateral KLMN is a parallelogram,
a trapezoid, a rectangle, or none of these.
e. Take it further: Prove a fact about midpoint quadrilaterals in general.
7. Graph quadrilateral GHIJ with G( 2 , 4), H(3, 3), I( − 2 , –4), and J(–3, –3).
How many right angles does it have? Prove your answer.
8. Using slopes, prove that every rectangle must be a parallelogram.
Hints:
•
•
•
Let m1, m2, m3, and m4 stand for the slopes of the four sides (going consecutively around
the rectangle).
We know that m1 · m2 = –1, m2 · m3 = –1, m3 · m4 = –1, and m4 · m1 = –1. (How do we
know this?)
What you need to prove is that m1 = m3 and m2 = m4.
page 3
Selected answers for “Coordinate geometry: working with slopes”
Notation used throughout: mXY stands for the slope of the line through points X and Y.
1 and 3.
D(–4, 3), E(–2,6), F(4,2)
mAB = mDE = 3/2
mAC = mDF = –1/8
mBC = mEF = –2/3
The slopes show that AB and
BC are perpendicular, so ΔABC
is a right triangle. (ΔDEF is too.)
Each side of the ΔDEF has the
same slope as a side of ΔABC.
2. In general, the midpoint is
⎛ x1 + x 2 y1 + y 2 ⎞
,
⎜
⎟ .
2 ⎠
⎝ 2
4. WXYZ is both a parallelogram and a rectangle, because the slopes of the sides are
mWX = mYZ = 5, mXY = mZW = –1/5.
5. STUV is a trapezoid with two right angles, because the slopes of the sides are
mST = 2, mTU = –1/2, mUV = 3/4, mVS = –1/2, which means that TU and VS are parallel
and there are right angles at S and T.
6. OPQR is a trapezoid because mOP = mQR = 2/3.
K(5, 2), L(6.5, 6), M(3.5, 7), N(2,3).
KLMN is a parallelogram because
mKN = mLM = –1/3, mKL = mMN = 4/1.5 or 8/3.
7. All 4 angles of GHIJ are right angles. Here’s how
to prove one of them; the others are similar.
mGH =
1
−3+ 2
mGH · mHI =
7
and mHI =
1
−3+ 2
(−3 + 2 )(3 + 2 )
=
·
7
3+ 2
7
3+ 2
7
.
=
−9−3 2 +3 2 + 2
=
7
= −1
−7
which proves that GH and HI are perpendicular at H.
8. Take m1 · m2 = m2 · m3 and divide by m2 to get m1 = m3. Show m2 = m4 similarly.