Lesson 4 Prove It Right 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 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 $A B C D$ a parallelogram? Explain how you know.

b.

Is $E F G H$ a parallelogram? Explain how you know.

2.

a.

Is $A B C D$ a rectangle? Explain how you know.

b.

Is $E F G H$ a rectangle? Explain how you know.

c.

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

3.

a.

Is $A B C D$ a rhombus? Explain how you know.

b.

Is $E F G H$ 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 $A B$ and side $B C$ of the triangle. Label these midpoints $M$ and $N$ . What relationship exists between segment $M N$ and side $A C$ of the triangle? Explain how you know.

b.

Now find the point $\frac{1}{3}$ of the distance from $A$ to $B$ and $\frac{1}{3}$ of the distance from $C$ to $B$ in the triangle. Label these points $R$ and $S$ . What relationship exists between segment $R S$ and side $A C$ of the triangle? Explain how you know.

c.

Find the area of triangle $A B C$ . (Hint: Use $\overset{―}{A C}$ as the base.)

Ready for More?

Find the midpoints of each of the sides of quadrilateral $A B C D$ and label the midpoints $M$ , $N$ , $P$ , and $Q$ . Figure $M N P Q$ 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 $4$ 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 $f (x)$ for the given domain values. Write $x$ and $f (x)$ as an ordered pair.

$f (x) = 2 x^{2} - 5 x$

$x$	$f (x)$	$(x, f (x))$
$- 2$
$- 1$
$0$
$1$
$2$

2.

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