geometry b - unit 1: coordinate geometry: slope and midpoints lessons 1-5

lesson 4

the midpoint formula

which reason justifies statement 3?

lines with equal slopes are parallel.

ab has endpoints at a(2, 6) and b(6, 8) what are the coordinates of the midpoint?

(4,7)

line segment ab has endpoints a(1, 8) and b(7, −4). what are the coordinates of the point located 5/6 of the way from a to b?

(6, -2)

what are the coordinates of point i?

(7, 4)

which equation in slope-intercept form represents a line that passes through the point (6, -1) and is perpendicular to the line y = 2x - 7?

y = -1/2x + 2

which information about the coordinates of g and h will prove that gh is a median of △abc?

g(3, -4) and h(8, 1)

ad has endpoints at a(4, −8) and b(−6,−2). what are the coordinates of the midpoint of ab?

(-1, -5)

the segment shown is half of ab, where b(−5, 1) is one endpoint of the segment and m(−3, 3) is the midpoint of the segment. what are the coordinates of point a?

(-1, 5)

cd has endpoints c and d, with c at coordinates (5, 8). cd has midpoint M at (3, 9). what are the coordinates of point d?

(1, 10)

use the diagram to answer the question. if cyrus knows that abc ~ edc and ab || ed, how can he prove that like m passing through ab has the same slope as line l passing through ed?

cyrus can use properties of similar triangles to show that ac/ec = bc/dc. then, by the multiplication property of equality, ac/bc = ec/dc. by the slope formula, the slope of line m = -ac/bc and the slope of line l = -ec/dc. thus, the slopes are equal.

lesson 3

coordinate proofs with slope

lesson 1

slope and parallel lines

4. match each numbered statement with the correct reason.

1. given 2. slope formula 3. multiply 4. if the product of the slopes of two nonvertical lines is −1 the lines are perpendicular. 5. perpendicular lines intersect to form right angles. 6. definition of right triangle

1. match each numbered statement with the correct reason.

1. given 2. slope formula 3. two distinct lines are parallel if and only if they have equal slopes. 4. exactly one pair of opposite sides of a trapezoid is parallel.

use de and fg to answer the question. is de perpendicular to fg? why or why not?

no, because the product of the slopes is not -1.

which information about the coordinates of d will prove that ad is a median of △abc?

d(1, 4)

which method will allow you to show ac and bd bisect each other?

find the midpoints of ac and bd using the midpoint formula. if the midpoints have the same coordinates, then the segments intersect at their midpoints.

line l contains the points a(-2, 2) and b(-1, 0), and line k contains the points c(0, 4) and d(1, 2) are lines l and k, parallel? justify your response.

yes, both lines have a slope of -2, so they are parallel.

line segment ab has endpoints a(−5, −8) and b(2, 6). what are the coordinates of the point that partitions ab according to the part-to-part ratio 4:3?

(-1, 0)

line segment ba has endpoints b(−3, 0) and a(5, 4). what are the coordinates of the point located 1/4 of the way from b to a?

(-1, 1)

line segment ba has endpoints b(−6, 1) and a(4, 6). what are the coordinates of the point that partitions ba according to the part-to-part ratio 2:3?

(-2, 3)

line segment ab has endpoints a(−1, 6) and b(5, −6). what are the coordinates of the point that partitions ab according to the part-to-part ratio 1:5?

(0, 4)

match each of the four lines with a line that is perpendicular to it.

y = 5/6x - 7 : y = -6/5x + 1 y = -5/6x - 8 : y = 6/5x - 5 y = -7/4x - 1 : y = 4/7x + 9 y = 7/4x - 2 : y = -4/7x + 2

lesson 2

slope and perpendicular lines

6. match each numbered statement with the correct reason.

1. given 2. slope formula 3. multiply 4. if the product of the slopes of two nonvertical lines is −1, the lines are perpendicular. 5. perpendicular lines intersect to form right angles. 6. definition of a right triangle

use the diagram and given information to answer the question. what description shows that lines l and m have slopes that are opposite reciprocals?

the triangles are similar, so ab/bc = cd/de. the slope of l = ab/bc and the slope of m = -de/cd. since cd/de and -de/cd are opposite reciprocals, lines l and m have slopes that are opposite reciprocals.

use the equation and given point to answer the questions below. select two answers: one for a and one for b.

a: -2 b: -10

which information about the coordinates of p will prove that fp is a median of △def?

p(0.5, 5)

which equation in slope-intercept form represents a line that passes through the point (2, 3) and is perpendicular to the line y - 9 = 2/3(x + 7)?

y = -3/2x + 6

what is the equation of the lines that passes through the given point and is perpendicular to the given line? point: (1, 1) line: y = 1/5x + 4/5

y = -5x + 6

for steps 4 through 8, match each numbered statement with the correct reason.

4. both pairs of opposite sides of a parallelogram are parallel. 5. multiply 6. if the product of the slopes of two nonvertical lines is −1, the lines are perpendicular. 7. perpendicular lines intersect to form right angles. 8. if a quadrilateral has four right angles, then it is a rectangle.

which equation in slope-intercept form represents a line that is parallel to y = -4x - 5 and passes through the point (0, 0)?

y = -4x

line segment ab has endpoints a(1, 8) and b(7, −4). what are the coordinates of the point located 1/6 of the way from a to b?

(2, 6)

use the following information to complete the problem. given: quadrilateral abcd has vertices at a(3, 6), b(2, 2), c(−5, 4), and d(−4, 8). prove: quadrilateral abcd is a parallelogram. match each numbered statement with the correct reason.

1. given 2. midpoint formula 3. if two lines intersect, then their intersection is exactly one point. 4. definition of midpoint 5. definition of bisect 6. if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

which plan for a proof for this problem will show that quadrilateral abdc is a parallelogram?

use the midpoint formula to find the midpoint of ad and bc. show that the midpoint for both segments is the same. by the definition of midpoint and bisect, ad and bc bisect each other. since the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

point b is at (3, 1). for which segment is b a midpoint?

ac with endpoints at a(1, -1) and c(5, 3)

allana knows that xyz ~ uvz and xy || uv. what segment relationships can allana use to prove that the slope of line l is equal to the slope of line m?

allana can show that uz/xz = vz/yz using the properties of similar triangles. then, uz/vz = xz/yz by the multiplicative property of equality. so, the slopes are equal by the slope formula.

lesson 5

splitting segments

which equation in slope-intercept form represents a line that is parallel to y = 1/2x - 2 and passes through the point (-8, 1)?

y = 1/2x + 5

which equation in slope-intercept form represents a line that passes through the point (2, 3) and is parallel to the line y - 9 = 2/3(x + 7)?

y = 2/3x + 5/3

which equation in slope-intercept form represents a line that passes through the point (5, -1) and is parallel to the line y = 2x - 7?

y = 2x - 11

use ab and cd to answer the question. ab contains the points a(2, 1) and b(3, 4) cd contains the points c(-2, -1) and d(1, -2) is ab perpendicular to cd? why or why not?

yes, because the product of the slopes is -1.

line l contains the points a(2, 2) and b(3, 6). line k contains the points c(0, 5) and d(1, 9). are lines l and k parallel? justify your response.

yes, lines l and k are parallel because the slopes are equal.

use the following information and diagram to complete the problem match each numbered statement with the correct reason.

1. given 2. midpoint formula 3. if two points have the same coordinates, then they are the same point. 4. definition of midsegment

match each numbered statement with the correct reason.

1. given 2. midpoint formula 3. it two lines intersect, then their intersection is exactly one point. 4. definition of midpoint 5. definition of bisect 6. if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

3. match each numbered statement with the correct reason.

1. given 2. slope formula 3. multiply 4. if the product of the slopes of two nonvertical lines is −1 the lines are perpendicular. 5. perpendicular lines intersect to form right angles. 6. if a quadrilateral has four right angles, then it is a rectangle.

examine the following lines and answer the question. given that line a is parallel to line b and the slope of line a is 2, what is the slope of line b?

2

5. match each numbered statement with the correct reason.

1. given 2. slope formula 3. lines with equal slopes are parallel 4. both pairs of opposite sides of a parallelogram are parallel.

examine the two distinct lines defined by the following two equations in slope-intercept form. are lines l and k parallel? justify your response.

yes lines l and k are parallel because their slopes are equal.

2. match each numbered statement with the correct reason.

1. given 2. slope formula 3. lines with equal slopes are parallel. 4. both sets of opposite sides of a parallelogram are parallel.

use the diagram and given information to answer the question. using the given information, how can it be proved that lines l and m have slopes that are opposite reciprocals?

since similar triangles have proportional sides, use the proportion bc/ab = de/cd. show that the slope of l = de/cd and the slope of m = -ab/bc. then show that bc/ab(-ab/bc) = -1, so the slopes of l and m are opposite reciprocals.