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  1. 1-5 Using Formulas in Geometry Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Warm Up Evaluate. Round to the nearest hundredth. 1. 122 2. 7.62 3. 4. 5. 32() 6. (3)2 144 57.76 8 7.35 28.27 88.83

  3. Objective Apply formulas for perimeter, area, and circumference.

  4. Vocabulary perimeter diameter area radius base circumference height pi

  5. The perimeterP of a plane figure is the sum of the side lengths of the figure. The areaA of a plane figure is the number of non-overlapping square units of a given size that exactly cover the figure.

  6. The basebcan be any side of a triangle. The heighthis a segment from a vertex that forms a right angle with a line containing the base. The height may be a side of the triangle or in the interior or the exterior of the triangle.

  7. Remember! Perimeter is expressed in linear units, such as inches (in.) or meters (m). Area is expressed in square units, such as square centimeters (cm2).

  8. Example 1A: Finding Perimeter and Area Find the perimeter and area of each figure. = 2(6) + 2(4) = 12 + 8 = 20 in. = (6)(4) = 24 in2

  9. Example 1B: Finding Perimeter and Area Find the perimeter and area of each figure. P = a + b + c = (x + 4) + 6 + 5x = 6x + 10 = 3x + 12

  10. Check It Out! Example 1 Find the perimeter and area of a square with s = 3.5 in. P = 4s A = s2 P = 4(3.5) A = (3.5)2 P = 14 in. A = 12.25 in2

  11. Example 2: Crafts Application The Queens Quilt block includes 12 blue triangles. The base and height of each triangle are about 4 in. Find the approximate amount of fabric used to make the 12 triangles. The area of one triangle is The total area of the 12 triangles is 12(8) = 96 in2.

  12. Find the amount of fabric used to make four rectangles. Each rectangle has a length of in. and a width of in. The area of one rectangle is The amount of fabric to make four rectangles is Check It Out! Example 2

  13. In a circle a diameter is a segment that passes through the center of the circle and whose endpoints are on a circle. A radius of a circle is a segment whose endpoints are the center of the circle and a point on the circle. The circumference of a circle is the distance around the circle.

  14. The ratio of a circle’s circumference to its diameter is the same for all circles. This ratio is represented by the Greek letter  (pi). The value of  is irrational. Pi is often approximated as 3.14 or .

  15. Example 3: Finding the Circumference and Area of a Circle Find the circumference and area of a circle with radius 8 cm. Use the key on your calculator. Then round the answer to the nearest tenth.  50.3 cm  201.1 cm2

  16. Check It Out! Example 3 Find the circumference and area of a circle with radius 14m.  88.0 m  615.8 m2

  17. Objectives Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points.

  18. Vocabulary coordinate plane leg hypotenuse

  19. A coordinate planeis a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis) . The location, or coordinates, of a point are given by an ordered pair (x, y).

  20. You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.

  21. Ordered pairs are used to locate points in a coordinate plane. y-axis (vertical axis) 5 5 -5 x-axis (horizontal axis) -5 origin (0,0)

  22. In an ordered pair, the first number is the x-coordinate. The second number is the y-coordinate.Graph. (-3, 2) 5 • 5 -5 -5

  23. What is the ordered pair for A? 5 • A • (3, 1) • (1, 3) • (-3, 1) • (3, -1) 5 -5 -5

  24. What is the ordered pair for B? 5 • (3, 2) • (-2, 3) • (-3, -2) • (3, -2) 5 -5 • B -5

  25. What is the ordered pair for C? 5 • (0, -4) • (-4, 0) • (0, 4) • (4, 0) 5 -5 • C -5

  26. What is the ordered pair for D? 5 • (-1, -6) • (-6, -1) • (-6, 1) • (6, -1) 5 -5 • D -5

  27. Write the ordered pairs that name points A, B, C, and D. A = (1, 3) B = (3, -2) C = (0, -4) D = (-6, -1) 5 • A 5 -5 • D • B • C -5

  28. The x-axis and y-axis separate the coordinate plane into four regions, called quadrants. II (-, +) I (+, +) III (-, -) IV (+, -)

  29. Name the quadrant in which each point is located(-5, 4) • I • II • III • IV • None – x-axis • None – y-axis

  30. Name the quadrant in which each point is located(-2, -7) • I • II • III • IV • None – x-axis • None – y-axis

  31. Helpful Hint To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane.

  32. Example 1: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7). = (–5, 5)

  33. Check It Out! Example 1 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).

  34. Step 2 Use the Midpoint Formula: Example 2: Finding the Coordinates of an Endpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y).

  35. – 2 – 7 –2 –7 Example 2 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by 2. 12 = 2 + x Simplify. 2 = 7 + y Subtract. –5 = y 10 = x Simplify. The coordinates of Y are (10, –5).

  36. S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T. Step 2 Use the Midpoint Formula: Check It Out! Example 2 Step 1 Let the coordinates of T equal (x, y).

  37. + 1 + 1 + 6 +6 Check It Out! Example 2 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by 2. –2 = –6 + x Simplify. 2 = –1 + y Add. 4 = x Simplify. 3 = y The coordinates of T are (4, 3).

  38. Warm UP Manuel is out for a jog. The thick lines on the grid are jogging paths. He is on his way home and is at D. His home is at E. Each unit on the grid is 1 mile. 1. Name the coordinates of D. 2. Find how many miles Manuel will jog if he goes straight to the x-axis. 3. Find how many miles Manuel will jog if he stays on the jogging paths all the way home.

  39. The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane.

  40. Find FG and JK. Then determine whether FG  JK. Example 3: Using the Distance Formula Step 1 Find the coordinates of each point. F(1, 2), G(5, 5), J(–4, 0), K(–1, –3)

  41. Example 3 Continued Step 2 Use the Distance Formula.

  42. Find EF and GH. Then determine if EF  GH. Check It Out! Example 3 Step 1 Find the coordinates of each point. E(–2, 1), F(–5, 5), G(–1, –2), H(3, 1)

  43. Check It Out! Example 3 Continued Step 2 Use the Distance Formula.

  44. You can also use the Pythagorean Theorem to find the distance between two points in a coordinate plane. You will learn more about the Pythagorean Theorem in Chapter 5. In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c.

  45. Example 4: Finding Distances in the Coordinate Plane Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(–2, –5).

  46. Example 4 Continued Method 1 Use the Distance Formula. Substitute the values for the coordinates of D and E into the Distance Formula.

  47. Example 4 Continued Method 2 Use the Pythagorean Theorem. Count the units for sides a and b. a = 5 and b = 9. c2 =a2 + b2 =52 + 92 =25 + 81 =106 c =10.3