Lesson 4Prove It RightPractice 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 lesson 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.

a.

Is a parallelogram? Explain how you know.

b.

Is a parallelogram? Explain how you know.

2.

a.

Is a rectangle? Explain how you know.

b.

Is a rectangle? Explain how you know.

c.

Find the perimeter of each of the figures in the diagram.

3.

a.

Is a rhombus? Explain how you know.

b.

Is a rhombus? Explain how you know.

c.

Find the area of each of the figures in the diagram.

4.

a.

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.

b.

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.

c.

Find the area of triangle . (Hint: Use as the base.)

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?

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 points on a coordinate plane formed the vertices of a parallelogram, rectangle, rhombus, or square. We also found the perimeter and area of figures defined by the coordinates of their vertices.

Retrieval

1.

Find the value of for the given domain values. Write and as an ordered pair.

2.

Find the perimeter for the triangle and the semicircle on the grid.