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Transcript
Name
March 10, 2016
Math 2 problem set
page 1
Coordinate geometry: using 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.
Exploration: perpendicular lines
Directions: Each picture shows a pair of perpendicular lines. Find the slopes of the lines.
Generalize: When you have a pair of perpendicular lines, how are the slopes related?
Name
March 10, 2016
Math 2 problem set
page 2
Conclusions and reference information
Ways to find slope
slope 
y  y1
rise
, slope  2
run
x2  x1
Linear equation forms
slope-intercept form: y = mx + b
point-slope form:
y = m(x – h) + k
m is the slope and the y-intercept is (0, b)
m is the slope and (h, k) is a point of the line
Parallel lines and perpendicular lines
algebraic meanings
Two lines are parallel if their slopes are equal.
Two lines are perpendicular if their slopes are
opposite reciprocals of each other.
geometric meanings
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 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 using averages. 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 midpoint triangle DEF.
d. How do the slopes for the midpoint triangle DEF compare to the slopes for triangle ABC?
Name
March 10, 2016
Math 2 problem set
page 3
Definitions of special types of quadrilaterals
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. The lines with these equations form a quadrilateral:
y=
2
5
x – 3,
y = – 52 x + 6,
y=
2
5
(x – 2) + 4,
y=
5
2
(x + 3) – 1
a. Graph the lines on your calculator to help you visualize the quadrilateral.
b. What are the slopes of the four sides?
c. What special type of quadrilateral is it? Tell how you know.
d. How many right angles does this quadrilateral have? Tell how you know.
5. 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.
6. Consider quadrilateral STUV with vertices S(0, 2), T(2, 6), U(6, 4), and V(2, 1).
a. Which angles of STUV are right angles? Justify your answer using slopes.
b. Prove that STUV is a trapezoid.
7. 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.
Name
March 10, 2016
Math 2 problem set
page 4
Problem answers (partial)
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 ABC
is a right triangle. (DEF is too.)
Each side of the DEF has the
same slope as a side of ABC.
2. In general, the midpoint is
 x1  x 2 y1  y 2 
,

.
2 
 2
4. trapezoid with two right angles
5. WXYZ is both a parallelogram and a rectangle, because the slopes of the sides are
mWX = mYZ = 5, mXY = mZW = –1/5.
6. 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.
7. 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.