Coordinate Geometry5-4

Coordinate Geometry5-4

Warm UpComplete each statement.

1. A triangle with an angle that measures 90 is a_______ triangle.right

trapezoid

parallelogram

square

2. A quadrilateral with exactly one pair of parallel sides is a ____________.

3. A quadrilateral with two pairs of parallel sides is a ____________.

4. A rhombus with 4 right angles is a ____________.

Coordinate Geometry5-4

Problem of the Day

What type of polygon am I? My opposite angles have equal measure. I do not have a right angle. All my sides are congruent. rhombus

Coordinate Geometry5-4

Learn to identify polygons and midpoints of segments in the coordinate plane.

Coordinate Geometry5-4

Vocabulary

midpoint

Coordinate Geometry5-4

In computer graphics, a coordinate system is used to create images, from simple geometric figures to realistic figures used in movies.

Properties of the coordinate plane can be used to find information about figures in the plane, such as whether lines in the plane are parallel.

Coordinate Geometry5-4

Additional Example 1A: Using Coordinates to Classify Quadrilaterals

Graph the polygons with the given vertices. Give the most specific name for the polygon.

A(0, –2), B(3, 3), C(3, –2)

Step 1: Classify the polygon by its angles.

C is a right angle, so triangle ACB is a right triangle.

Coordinate Geometry5-4

Additional Example 1A Continued

Step 2: Classify the triangle by its sides.

Find the length of each side.

AC = |3 – 0| = 3

BC = |–2 – 3| = 5

The triangle has no congruent sides, so it is scalene. Triangle ACB is a right scalene triangle.

Coordinate Geometry5-4

Additional Example 1B: Using Coordinates to Classify Quadrilaterals

Graph the polygons with the given vertices. Give the most specific name for the polygon.

L(2, 1), M(5, 1), N(5, –1), P(2, –1)

LM and PN are parallel.

LP and MN are parallel.The quadrilateral is a rectangle because it has two sets of parallel sides and the sets of parallel sides are different lengths.

Examine the sides of the quadrilateral.

Coordinate Geometry5-4Check It Out: Example 1

Graph the polygons with the given vertices. Give the most specific name for the polygon.

Q(–2, 3), R(1, 5), S(1, –2), T(–2, –4)

QT and RS are parallel.

QR and TS are parallel.The quadrilateral is a parallelogram because it has two sets of parallel sides and the sets of parallel sides do not form 90 angles.

Examine the sides of the quadrilateral.

Coordinate Geometry5-4

Additional Example 2A: Finding the Coordinates of a Missing Vertex

Find the coordinates of each missing vertex.

Triangle ABC has a right angle at C and AC = 2. Find one set of possible coordinates for A.

Since BC is horizontal, AC must be vertical for the triangle to have a right angle at C.(4, 0) or (4, –4)

Coordinate Geometry5-4

Additional Example 2B: Finding the Coordinates of a Missing Vertex

Find the coordinates of each missing vertex.

Quadrilateral LMNP is a parallelogram. Find the coordinates of N.

MN must be vertical and the same length as LP.

(1, 0)

Coordinate Geometry5-4

Check It Out: Example 2

Find the coordinates of each missing vertex.

Triangle ABC has a right angle at B and AB = 3. Find one set of possible coordinates for A.

Since BC is horizontal, AB must be vertical for the triangle to have a right angle at B.(1, 1) or (1, –5)

Coordinate Geometry5-4

The midpoint of a segment is the point that divides the segment into two congruent segments.

Coordinate Geometry5-4

Additional Example 3: Finding the Coordinates of a Midpoint

Find the coordinates of midpoint AB.

The coordinates of the midpoint M is

Use the formula.

The endpoints are A (0, 2) and B(3, 5).

Simplify.

Coordinate Geometry5-4

Check It Out: Example 3

Find the coordinates of midpoint LM.

Use the formula.

The endpoints are L (–1, 3) and M(2, 1).

Simplify.

The coordinates of the midpoint M is

Coordinate Geometry5-4

Lesson Quiz: Part I

1. Graph the polygon with vertices A(0, 1),

B(1, 3), C(3, 3), and D(5, 1).

trapezoid

Coordinate Geometry5-4

Lesson Quiz: Part II

2. Quadrilateral LMNP is a parallelogram.

Find the coordinates of P.

3. Find the coordinates of the midpoint MN.

(4, 1)

(5, 2.5)

Coordinate Geometry5-4

1. Graph the polygon with vertices A(1, 1), B(1, 3), C(3, 3), and D(3, 1). Give the most specific name for the polygon.

A. square

B. rectangle

C. parallelogram

D. trapezoid

Lesson Quiz for Student Response Systems

Coordinate Geometry5-4

2. Find the coordinates of the midpoint CD.

A. (2, 2)

B. (2, 1)

C. (4, 2)

D. (4, 1)

Lesson Quiz for Student Response Systems