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 9-3, 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 1A: 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 = The opposite sides are parallel, so EFGH is a parallelogram. slope of HE =

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, 1), B(2, 4), C(4, 1), and D(–2, –2). 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.

Example 4: Problem Solving Application Show that the area does not change when the pieces are rearranged.

Understand the Problem Example 4 Continued 1 Understand the Problem The parts of the puzzle appear to form two trapezoids with the same bases and height that contain the same shapes, but one appears to have an area that is larger by one square unit.

Example 4 Continued 2 Make a Plan Find the areas of the shapes that make up each figure. If the corresponding areas are the same, then both figures have the same area by the Area Addition Postulate. To explain why the area appears to increase, consider the assumptions being made about the figure. Each figure is assumed to be a trapezoid with bases of 2 and 4 units and a height of 9 units. Both figures are divided into several smaller shapes.

Example 4 Continued Solve 3 Find the area of each shape. Left figure Right figure top triangle: top triangle: top rectangle: top rectangle: A = bh = 2(5) = 10 units2 A = bh = 2(5) = 10 units2

Example 4 Continued Solve 3 Find the area of each shape. Left figure Right figure bottom triangle: bottom triangle: bottom rectangle: bottom rectangle: A = bh = 3(4) = 12 units2 A = bh = 3(4) = 12 units2

Example 4 Continued Solve 3 The areas are the same. Both figures have an area of 2.5 + 10 + 2 + 12 + = 26.5 units2. If the figures were trapezoids, their areas would be A = (2 + 4)(9) = 27 units2. By the Area Addition Postulate, the area is only 26.5 units2, so the figures must not be trapezoids. Each figure is a pentagon whose shape is very close to a trapezoid.

Example 4 Continued Look Back 4 The slope of the hypotenuse of the smaller triangle is 4. The slope of the hypotenuse of the larger triangle is 5. Since the slopes are unequal, the hypotenuses do not form a straight line. This means the overall shapes are not trapezoids.

Check It Out! Example 4 Create a figure and divide it into pieces so that the area of the figure appears to increase when the pieces are rearranged. Check the students' work.

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 4. Show that the two composite figures cover the same area. For both figures, A = 3 + 1 + 2 = 6 units2.