Geometry HS Mathematics Unit: 02 Lesson: 01
Slope, Distance, Midpoint KEY Line segments in a coordinate plane can be analyzed by finding various characteristics of the line including slope, length, and midpoint. These values are useful in applications and coordinate proofs. Slope = • Parallel lines –slopes are equal. • Perpendicular lines –slopes are negative reciprocals. • Some lines are neither parallel nor perpendicular.
(x 1 , y 1 ) a n d (x 2 , y 2 ) Length of a segment is the distance between
.
Distance formula
(x 1 , y 1 ) a n d (x 2 , y 2 ) Midpoint of a segment between Midpoint formula Examples: 1. Given A(9, -5) and B(-6, 12) a. Find the slope. -17/15 b. Find AB. Approx. 22.7 AB c. Find Q, the midpoint of .
(3/2, 7/2)
d. Find the equation of the line that passes through (SF: 17x + 15y = 78) 2. Given R(-2, 5) and T(4, 1) a. Find the slope. -2/3 b. Find RT. Approx. 7.2 RT c. Find S, the midpoint of .
.
y = (-17/15)x + 26/5
.
y = (-2/3)x + (11/3)
(1, 3)
d. Find the equation of the line that passes through (SF: 2x + 3y = 11)
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AB
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RT
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Geometry HS Mathematics Unit: 02 Lesson: 01
Slope, Distance, Midpoint KEY Comparing Parallelism in Euclidean and non-Euclidean Geometries Parallel Postulate (Euclidean Geometry) – There is exactly one line in a plane that can be drawn parallel to another line through a point not on the line. If the Parallel Postulate is considered false, then one of the following assumptions must be considered true. Assumption 1: Through a given point not on a line, there are no lines parallel to the given line. Assumption 2: Through a given point not on a line, there is more than one line parallel to the given line. Assumption 1 applies to spherical geometry. Assume that the following figure is a sphere. Lines are represented by curves on the great circles of the sphere. This is a sphere and in spherical geometry, great circles represent lines. Note that all great circles intersect. Therefore there are no parallel “lines”.
p t v s
Assumption 2 applies to hyperbolic geometry. Assume that the following figure is a concave disk. Lines appear to be curves on the surface of the disk, although if flattened into a plane, they would be linear. This is not a sphere, but is the inside of a curved disk. Note that p is parallel to s, t, and v, since they do not intersect. There are an infinite number of curves that could be drawn through the point and not intersect p.
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Geometry HS Mathematics Unit: 02 Lesson: 01
Slope, Distance, Midpoint KEY Guided Practice 1. Use the coordinate plane to answer questions a – f.
a. Given A(0, -8), B(5, 9), C(7, -6), D(-8, 6), E(3, 0), F(-4, -2), G(-6, 2), label the points on the graph.
b. Find the slope of m of
suur BE
= 9/2, m of
c. Find the slope of m of
suur FE
suur BE
suur FE
= 2/7, m of
d. Find the length of
AE
e. Find the midpoint of
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sand uur EG
sand uur AC
suur EG
. What does the slope indicate about these lines?
= -2/9 Lines are perpendicular.
suur AC
. What does the slope indicate about these lines?
= 2/7 Lines are parallel.
. Approx. 8.5
CF
. (3/2, -4)
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Geometry HS Mathematics Unit: 02 Lesson: 01
f.
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Find the length and midpoint of
CD
. Approx. 19.2 and (-1/2, 0)
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Geometry HS Mathematics Unit: 02 Lesson: 01
Slope, Distance, Midpoint KEY Practice Problems 1. A square is represented by the points A(1, -2), B(-3, 2), C(-7, -2), and D(-3, -6). a. Find the equation of the line passing through each side of square ABCD. Line AB, y = -x – 1; x + y = -1 Line BC, y = x + 5; x – y = -5 Line CD, y = -x – 9; x + y = -9 Line DA, y = x – 3; x – y = 3 b. Find the slope of each side of square ABCD. Slope AB, m = -1 Slope BC, m = 1 Slope CD, m = -1 Slope DA, m = 1 c. Determine which sides are perpendicular and which sides are parallel. Line AB is parallel to line CD. Line BC is parallel to line DA. Line AB is perpendicular to lines BC and DA. Line CD is perpendicular to lines BC and DA. d. Calculate the length of each line segment that makes up square ABCD. 32 4 2 Line AB, d = or , 5.7 32 4 2 Line BC, d = or , 5.7 32 4 2 Line CD, d = or , 5.7 32 4 2 Line DA, d = or , 5.7 e. Construct a graph as an alternate representation to verify your conclusions.
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Geometry HS Mathematics Unit: 02 Lesson: 01
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Geometry HS Mathematics Unit: 02 Lesson: 01
Slope, Distance, Midpoint KEY Use the coordinate plane to answer the questions below.
2. Given H(-1, -3), I(0, 6), J(6, -3), K(-6, 0), L(-4, 4), M(1, 2), N(3, 7), label the points on the graph.
3. Find the suurslope of m of
MN
m of
suur LM
suur HK
suur MJ
and . What does the slope indicate about these lines? suur LM m of = -2/5 They are perpendicular.
= 5/2
4. Find the slope of
suur HK
suur MN
= -3/5 m of
5. Find the length of
KI
6. Find the midpoint of
sand uur MJ
. What does the slope indicate about these lines?
= -1
. Approx. 8.5 KM
. (-5/2, 1)
7. Find the length and midpoint of
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They are neither parallel or perpendicular.
JL
. Approx. 12.2 (1, ½)
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Geometry HS Mathematics Unit: 02 Lesson: 01
Slope, Distance, Midpoint KEY Euclidean versus non-Euclidean Geometries 8. What comprises the surface of a sphere? Answers may vary. The surface area of the sphere
9. What is a “line” on a sphere? Answers may vary. A great circle of the sphere
10. Does the Parallel Postulate in Euclidean geometry hold in spherical geometry? Explain your reasoning. No it does not. Since all lines on a sphere are great circles and all great circles intersect in two places, there are no parallel lines in spherical geometry.
11. What comprises a surface in hyperbolic geometry? Answers may vary. The surface area of a concave disk, i.e. satellite dish
12. What is a “line” in hyperbolic geometry? Answers may vary. An arc on the disk
13. Does the Parallel Postulate in Euclidean geometry hold in hyperbolic geometry? Explain your reasoning. No it does not. An infinite number of arcs can be drawn through a point and not intersect another given arc.
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