Lesson 4 Prove It with Algebra Practice Understanding

Learning Focus

Prove quadrilaterals are parallelograms, rectangles, rhombi, or squares using coordinates.

Find the perimeter and area of a quadrilateral on the coordinate plane.

How do I use algebra to show that a quadrilateral is a parallelogram, a rectangle, a rhombus, or a square?

Open Up the Math: Launch, Explore, Discuss

In this task you need to use all the things you know about quadrilaterals, distance, and slope to prove that the shapes are parallelograms, rectangles, rhombi, or squares. Be systematic and be sure that you give all the evidence necessary to verify your claim.

1.

quadrilaterals ABCD and EFGH on a plane x–14–14–14–12–12–12–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414161616181818202020y–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414000A = (-10, 12)A = (-10, 12)A = (-10, 12)B = (-4, 12)B = (-4, 12)B = (-4, 12)D = (-12, 8)D = (-12, 8)D = (-12, 8)C = (-6, 8)C = (-6, 8)C = (-6, 8)E = (5, 2)E = (5, 2)E = (5, 2)F = (15, 0)F = (15, 0)F = (15, 0)H = (2, -6)H = (2, -6)H = (2, -6)G = (13, -9)G = (13, -9)G = (13, -9)

a.

Is a parallelogram? Explain how you know.

b.

Is a parallelogram? Explain how you know.

2.

quadrilaterals ABCD and EFGH on a plane x–16–16–16–14–14–14–12–12–12–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414161616181818202020222222242424262626y–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414161616181818000B = (2, 13)B = (2, 13)B = (2, 13)A = (-8, 13)A = (-8, 13)A = (-8, 13)C = (2, 9)C = (2, 9)C = (2, 9)D = (-8, 9)D = (-8, 9)D = (-8, 9)E = (6, 6)E = (6, 6)E = (6, 6)F = (14, 0)F = (14, 0)F = (14, 0)G = (7, -9)G = (7, -9)G = (7, -9)H = (-1, -3)H = (-1, -3)H = (-1, -3)

a.

Is a rectangle? Explain how you know.

b.

Is a rectangle? Explain how you know.

3.

quadrilaterals ABCD and EFGH on a plane x–12–12–12–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414y–6–6–6–4–4–4–2–2–2222444666888101010000A = (9, 8)A = (9, 8)A = (9, 8)B = (9, 2)B = (9, 2)B = (9, 2)C = (3, 3)C = (3, 3)C = (3, 3)D = (3, 9)D = (3, 9)D = (3, 9)E = (-6, 3)E = (-6, 3)E = (-6, 3)F = (-4, -2)F = (-4, -2)F = (-4, -2)G = (-6, -6)G = (-6, -6)G = (-6, -6)H = (-8, -2)H = (-8, -2)H = (-8, -2)

a.

Is a rhombus? Explain how you know.

b.

Is a rhombus? Explain how you know.

4.

quadrilateral ABCD on a plane x–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111121212131313141414151515161616171717181818191919202020y–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111121212131313141414A = (7, 2)A = (7, 2)A = (7, 2)B = (14, 6)B = (14, 6)B = (14, 6)C = (10, 13)C = (10, 13)C = (10, 13)D = (3, 9)D = (3, 9)D = (3, 9)

a.

Is a square? Explain how you know.

b.

Find the perimeter and area of the quadrilateral.

5.

  1. Find the midpoint of side and side of the triangle. Label these midpoints and . What relationship exists between segment and side of the triangle? Explain how you know.

  2. Now find the point of the distance from to and of the distance from to in the triangle. Label these points and . What relationship exists between segment and side of the triangle? Explain how you know.

triangle ABC on a plane with points (3,9), (15,15), and (9,3) x555101010151515202020y555101010151515000A = (3, 9)A = (3, 9)A = (3, 9)B = (15, 15)B = (15, 15)B = (15, 15)C = (9, 3)C = (9, 3)C = (9, 3)

Ready for More?

Find the midpoints of each of the sides of quadrilateral and label the midpoints , , , and . Figure is what type of quadrilateral? How do you know?

quadrilateral ABCD on a plane x555101010151515202020y555101010151515000A = (2, 2)A = (2, 2)A = (2, 2)B = (6, 12)B = (6, 12)B = (6, 12)C = (14, 10)C = (14, 10)C = (14, 10)D = (16, 0)D = (16, 0)D = (16, 0)

Takeaways

Ways to use coordinates to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares:

Lesson Summary

In this lesson, we used the distance formula, the midpoint rule, and the properties of slopes of parallel and perpendicular lines to determine if a given set of 4 points on a coordinate plane formed the vertices of a parallelogram, rectangle, rhombus, or square.

Retrieval

1.

Find the values that will make the equations true.

a.

b.

2.

How many combinations of values for and do you think there are that would make the equation true? Explain.

3.

Find the perimeter of pentagon . Show the exact value and then round your answer to the nearest thousandths place.

Pentagon PQRST P(1,2) Q(2,4) R(3,5), S(5,4), T(4,1) x555y555000