Coordinate Geometry Proofs

Show your work for each problem.

1.
Which formula should you use to prove that two sides of a quadrilateral are equal in length?
1. a)Slope formula
2. b)Quadratic formula
3. c)Midpoint formula
4. d)Distance formula
Answer: Distance formula
The distance formula calculates the length of a segment between two points.
2.
Two lines are perpendicular if their slopes multiply to:
1. a)
2. b)
3. c)undefined
4. d)
Answer:
For two non-vertical, non-horizontal lines to be perpendicular, their slopes must be negative reciprocals: .
3.
Find the slope of the line passing through and .
Answer: 1
Slope .
4.
Find the distance between and .
Answer: 5
Distance .
5.
What does it mean if two lines have the same slope?
1. a)They are the same line
2. b)They are perpendicular
3. c)They are parallel
4. d)They intersect at the origin
Answer: They are parallel
Lines with equal slopes are parallel. They will never intersect (unless they are the same line).
6.
Find the midpoint of the segment with endpoints and . Enter as .
Answer: (4, 6)
Midpoint .
7.
Prove that with and is parallel to with and .
Answer: Both slopes equal 1, so the lines are parallel
1. Calculate the slope of : 1
2. Calculate the slope of : 1
3. Are the slopes equal? yes
4. Therefore, are and parallel? yes
8.
To prove that a quadrilateral is a rhombus using coordinates, what must you show?
1. a)All four sides have equal length
2. b)All angles are 90 degrees
3. c)Diagonals are equal
4. d)Opposite sides are parallel
Answer: All four sides have equal length
A rhombus is a quadrilateral with all four sides equal. Use the distance formula to verify all sides have the same length.
9.
Prove that lines with slopes and are perpendicular.
Answer: The product of slopes is -1, so the lines are perpendicular
1. What is ? Calculate -1
2. What must the product of slopes equal for perpendicular lines? -1
3. Since , are the lines perpendicular? yes
10.
Find the distance between and .
Answer: 10
.
11.
Verify that quadrilateral with , , , is a rectangle by showing adjacent sides are perpendicular.
Answer: ABCD is a rectangle
1. What is the slope of ? (horizontal line) 0
2. What is the slope of ? (vertical line) undefined
3. Are horizontal and vertical lines perpendicular? yes
4. Is a rectangle? yes
12.
Prove that quadrilateral with , , , is a parallelogram by showing diagonals bisect each other.
Answer: (4.5, 3)
1. What is the midpoint of diagonal ? (4.5, 3)
2. What is the midpoint of diagonal ? (4.5, 3)
3. Are the midpoints of both diagonals the same? yes
4. Therefore, is a parallelogram? yes
13.
If has slope , what slope must have to be perpendicular to ? Enter as a fraction.
Answer: -4/3
For perpendicular lines, . So , giving .
14.
Prove that quadrilateral with vertices , , , is a square.
Answer: WXYZ is a square
1. Calculate using distance formula: 5
2. Calculate : 5
3. Calculate : 5
4. Calculate : 5
5. Are all four sides equal? yes
6. Are adjacent sides perpendicular? (horizontal and vertical) yes
7. Is this shape a square? yes
15.
Which property distinguishes a rectangle from a general parallelogram?
1. a)Diagonals bisect each other
2. b)Adjacent sides are perpendicular
3. c)Opposite sides are equal
4. d)Opposite sides are parallel
Answer: Adjacent sides are perpendicular
A rectangle is a parallelogram with perpendicular adjacent sides (right angles). The other properties are shared by all parallelograms.