10-4 Perimeter and Area in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

Warm Up Use the slope formula to determine the slope of each line. 1. 2. 3. Simplify

Objective Find the perimeters and areas of figures in a coordinate plane.

In Lesson 10-3, you estimated the area of irregular shapes by drawing composite figures that approximated the irregular shapes and by using area formulas. Another method of estimating area is to use a grid and count the squares on the grid.

Example 1: Estimating Areas of Irregular Shapes in the Coordinate Plane Estimate the area of the irregular shape.

Example 1A Continued Method 1: Draw a composite figure that approximates the irregular shape and find the area of the composite figure. The area is approximately 4 + 5.5 + 2 + 3 + 3 + 4 + 1.5 + 1 + 6 = 30 units2.

Example 1A Continued Method 2: Count the number of squares inside the figure, estimating half squares. Use a  for a whole square and a for a half square. There are approximately 24 whole squares and 14 half squares, so the area is about

Check It Out! Example 1 Estimate the area of the irregular shape. There are approximately 33 whole squares and 9 half squares, so the area is about 38 units2.

Remember!

Example 2: Finding Perimeter and Area in the Coordinate Plane Draw and classify the polygon with vertices E(–1, –1), F(2, –2), G(–1, –4), and H(–4, –3). Find the perimeter and area of the polygon. Step 1 Draw the polygon.

Example 2 Continued Step 2 EFGH appears to be a parallelogram. To verify this, use slopes to show that opposite sides are parallel.

Example 2 Continued slope of EF = slope of GH = slope of FG = slope of HE = The opposite sides are parallel, so EFGH is a parallelogram.

Example 2 Continued Step 3 Since EFGH is a parallelogram, EF = GH, and FG = HE. Use the Distance Formula to find each side length. perimeter of EFGH:

Example 2 Continued To find the area of EFGH, draw a line to divide EFGH into two triangles. The base and height of each triangle is 3. The area of each triangle is The area of EFGH is 2(4.5) = 9 units2.

Check It Out! Example 2 Draw and classify the polygon with vertices H(–3, 4), J(2, 6), K(2, 1), and L(–3, –1). Find the perimeter and area of the polygon. Step 1 Draw the polygon.

Check It Out! Example 2 Continued Step 2 HJKL appears to be a parallelogram. To verify this, use slopes to show that opposite sides are parallel.

Check It Out! Example 2 Continued are vertical lines. The opposite sides are parallel, so HJKL is a parallelogram.

Check It Out! Example 2 Continued Step 3 Since HJKL is a parallelogram, HJ = KL, and JK = LH. Use the Distance Formula to find each side length. perimeter of EFGH:

Check It Out! Example 2 Continued To find the area of HJKL, draw a line to divide HJKL into two triangles. The base and height of each triangle is 3. The area of each triangle is The area of HJKL is 2(12.5) = 25 units2.

Example 3: Finding Areas in the Coordinate Plane by Subtracting Find the area of the polygon with vertices A(–4, 0), B(2, 3), C(4, 0), and D(–2, –3). Draw the polygon and close it in a rectangle. Area of rectangle: A = bh = 8(6)= 48 units2.

Example 3 Continued Area of triangles: The area of the polygon is 48 – 9 – 3 – 9 – 3 = 24 units2.

Check It Out! Example 3 Find the area of the polygon with vertices K(–2, 4), L(6, –2), M(4, –4), and N(–6, –2). Draw the polygon and close it in a rectangle. Area of rectangle: A = bh = 12(8)= 96 units2.

Check It Out! Example 3 Continued Area of triangles: a b d c The area of the polygon is 96 – 12 – 24 – 2 – 10 = 48 units2.

Lesson Quiz: Part I 1. Estimate the area of the irregular shape. 25.5 units2 2. Draw and classify the polygon with vertices L(–2, 1), M(–2, 3), N(0, 3), and P(1, 0). Find the perimeter and area of the polygon. Kite; P = 4 + 2√10 units; A = 6 units2

Lesson Quiz: Part II 3. Find the area of the polygon with vertices S(–1, –1), T(–2, 1), V(3, 2), and W(2, –2). A = 12 units2