1. In the figure, ABCD is an isosceles trapezoid with median EF. Find mD if mA = 110.
2. Find x if AD = 3x2 – 5 and BC = x2 + 27.
3. Find y if AC = 9(2y – 4) and
BD = 10y + 12.
4. Find EF if AB = 10 and CD = 32.
5. Find AB if AB = r + 18, CD = 6r + 9, and EF = 4r + 10.
• Position and label quadrilaterals for use in coordinate proofs.
• Prove theorems using coordinate proofs.
Positioning a Square
POSITIONING A RECTANGLE Position and label a rectangle with sides a and b units long on the coordinate plane.
The y-coordinate of B is 0 because the vertex is on the x-axis. Since the side length is a, the x-coordinate is a.
Let A, B, C, and D be vertices of a rectangle with sides a units long, and sides b units
long.
Place the square with vertex A at the origin, along the
positive x-axis, and along the positive y-axis. Label the vertices A, B, C, and D.
Positioning a Square
Sample answer:
D is on the y-axis so the x-coordinate is 0. Since the side length is b, the y-coordinate is b.
The x-coordinate of C is also a. The y-coordinate is 0 + b or b because the side is b units long.
A. A
B. B
C. C
D. D
Position and label a square with sides a units long on the coordinate plane. Which diagram would best achieve this?
A. B.
C. D.
Find Missing Coordinates
Name the missing coordinates for the isosceles trapezoid.
Answer: D(b, c)
The legs of an isosceles trapezoid are congruent and have opposite slopes. Point C is c units up and b units to the left of B. So, point D is c units up and b units to the right of A. Therefore, the x-coordinate of D is 0 + b, or b, and the y-coordinate of D is 0 + c, or c.
1. A
2. B
3. C
4. D
A. C(c, c)
B. C(a, c)
C. C(a + b, c)
D. C(b, c)
Name the missing coordinates for the parallelogram.
Coordinate Proof
Place a rhombus on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rectangle.The first step is to position a rhombus on the coordinate plane so that the origin is the midpoint of the diagonals and the diagonals are on the axes, as shown. Label the vertices to make computations as simple as possible.Given: ABCD is a rhombus as labeled. M, N, P, Q are
midpoints.
Prove: MNPQ is a rectangle.
Coordinate Proof
Proof:
By the Midpoint Formula, the coordinates of M, N, P, and Q are as follows.
Find the slopes of
Coordinate Proof
Coordinate Proof
A segment with slope 0 is perpendicular to a segment with undefined slope. Therefore, consecutive sides of this quadrilateral are perpendicular. MNPQ is, by definition, a rectangle.
Place an isosceles trapezoid on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rhombus.
Given: ABCD is an isosceles trapezoid. M, N, P, and Q are midpoints.
Prove: MNPQ is a rhombus.
The coordinates of M are (–3a, b); the coordinates of N
are (0, 0); the coordinates of P are (3a, b); the
coordinates of Q are (0, 2b).
Since opposite sides have equal slopes, opposite sides
are parallel and MNPQ is a parallelogram. The slope of
The slope of is undefined. So, the diagonals
are perpendicular. Thus, MNPQ is a rhombus.
Proof:
1. A
2. B
3. C
4. D
A.
B.
C.
D.
Which expression would be the lengths of the four sides of MNPQ?
Properties of Quadrilaterals
Proof:
Since have the same slope, they are parallel.
Write a coordinate proof to prove that the supports of a platform lift are parallel.
Prove:
Given: A(5, 0), B(10, 5), C(5, 10), D(0, 5)
A. A
B. B
C. C
D. D
A. slopes = 2
B. slopes = –4
C. slopes = 4
D. slopes = –2
Prove:
Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4)