Lesson 3 Prove It! 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.

Quadrilateral ABCD A(-10,12), B(-4,12), C(-6,8), and D(-12,8); Quadrilateral EFGH E(5,2), F(15,0), H(2,-6), G(13,-9). x–15–15–15–10–10–10–5–5–5555101010151515202020y–10–10–10–5–5–5555101010151515000(-10, 12)(-10, 12)(-10, 12)(-4, 12)(-4, 12)(-4, 12)(-6, 8)(-6, 8)(-6, 8)(-12, 8)(-12, 8)(-12, 8)(5, 2)(5, 2)(5, 2)(15, 0)(15, 0)(15, 0)(13, -9)(13, -9)(13, -9)(2, -6)(2, -6)(2, -6)

a.

Is a parallelogram? Explain how you know.

b.

Is a parallelogram? Explain how you know.

2.

Quadrilateral ABCD A(-8,13), B(2,13), C(-8,9), and D(-8,9); Quadrilateral EFGH E(6,6), F(14,0), H(-1,-3), G(7,-9). x–10–10–10–5–5–5555101010151515y–10–10–10–5–5–5555101010151515000

a.

Is a rectangle? Explain how you know.

b.

Is a rectangle? Explain how you know.

3.

Quadrilateral ABCD A(9,8), B(9,2), C(3,3), and D(3,9); Quadrilateral EFGH E(-6,3), F(-4,-2), H(-8,-2), G(-6,-6). x–10–10–10–5–5–5555101010y–5–5–5555000

a.

Is a rhombus? Explain how you know.

b.

Is a rhombus? Explain how you know.

4.

Quadrilateral ABCD A(7,2), B(14,6), C(10,13), D(3,9). x555101010151515202020y555101010

Is a square? Explain how you know.

Ready for More?

How can you show that a quadrilateral is a square with the fewest steps?

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 four 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