Areas of Trapezoids, Rhombuses, and Kites

1. 1. rectangle 2. a triangle 3.4. 5. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Check Skills You’ll Need (For help, go to Lesson 10-1.) Write the formula for the area of each type of figure. Find the area of each trapezoid by using the formulas for area of a rectangle and area of a triangle. Check Skills You’ll Need 10-2

2. 1.A = bh 2.A = bh 3. Draw a segment from S perpendicular to UT. This forms a rectangle and a triangle. The area A of the triangle is bh = (1)(2) = 1. The area A of the rectangle is bh = (4)(2) = 8. By Theorem 1-10, the area of a region is the sum of the area of the nonoverlapping parts. So, add the two areas: 1 + 8 = 9 units2. 1 2 1 2 1 2 Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Check Skills You’ll Need Solutions 10-2

3. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Check Skills You’ll Need Solutions (continued) 4. Draw two segments, one from M perpendicular to CB and the other from K perpendicular to CB. This forms two triangles and a rectangle between them. The area A of the triangle on the left is bh = (1)(2) = 1. The area A of the triangle on the right is bh = (2)(2) = 2. The area A of the rectangle is bh = (2)(2) = 4. By Theorem 1–10, the area of a region is the sum of the area of the nonoverlapping parts. So, add the three areas: 1 + 2 + 4 = 7 units2. 5. Draw two segments, one from A perpendicular to CD and the other from B perpendicular to CD. This forms two triangles and a rectangle between them. The area A of the triangle on the left is bh = (2)(3) = 3. The area A of the triangle on the right is bh = (3)(3) = 4.5. The area A of the rectangle is bh = (2)(3) = 6. By Theorem 1–10, the area of a region is the sum of the area of the nonoverlapping parts. So, add the three areas: 3 + 4.5 + 6 = 13.5 units2. 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 10-2

4. 1. Find the area of the parallelogram. 2. Find the area of XYZW with vertices X(–5, –3), Y(–2, 3), Z(2, 3) and W(–1, –3). 3. A parallelogram has 6-cm and 8-cm sides. The height corresponding to the 8-cm base is 4.5 cm. Find the height corresponding to the 6-cm base. 4. Find the area of RST. 5. A rectangular flag is divided into four regions by its diagonals. Two of the regions are shaded. Find the total area of the shaded regions. Areas of Parallelograms and Triangles Lesson 10-1 Lesson Quiz 150 ft2 15 m2 24 square units 187 in.2 6 cm 10-2

5. Textbook

7. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Notes The height of a trapezoid is the perpendicular distance hbetween the bases. 10-2

8. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Notes 10-2

9. A car window is shaped like the trapezoid shown. Find the area of the window. 1 2 A = h(b1 + b2) Area of a trapezoid 1 2 A = (18)(20 + 36) Substitute 18 for h, 20 for b1, and 36 for b2. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Additional Examples Real-World Connection A = 504 Simplify. The area of the car window is 504 in.2 Quick Check 10-2

10. Find the area of trapezoid ABCD. Draw an altitude from vertex B to DC that divides trapezoid ABCD into a rectangle and a right triangle. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Additional Examples Finding the Area Using a Right Triangle Because opposite sides of rectangle ABXD are congruent, DX = 11 ft and XC = 16 ft – 11 ft = 5 ft. 10-2

11. (continued) By the Pythagorean Theorem, BX2 + XC2 = BC2, so BX2 = 132 – 52 = 144. Taking the square root, BX = 12 ft. You may remember that 5, 12, 13 is a Pythagorean triple. 1 2 A = h(b1 + b2) Use the trapezoid area formula. 1 2 A = (12)(11 + 16) Substitute 12 for h, 11 for b1, and 16 for b2. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Additional Examples A = 162 Simplify. The area of trapezoid ABCD is 162 ft2. Quick Check 10-2

12. 1 2 A = d1d2Use the formula for the area of a kite. 1 2 A = (6)(5) Substitute 6 for d1 and 5 for d2. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Additional Examples Finding the Area of a Kite Find the area of kite XYZW. Find the lengths of the diagonals of kite XYZW. XZ = d1 = 3 + 3 = 6 and YW = d2 = 1 + 4 = 5 A = 15 Simplify. Quick Check The area of kite XYZW is 15 cm2. 10-2

13. To find the area, you need to know the lengths of both diagonals. Draw diagonal SU, and label the intersection of the diagonals point X. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Additional Examples Finding the Area of a Rhombus Find the area of rhombus RSTU. 10-2

14. (continued) SXT is a right triangle because the diagonals of a rhombus are perpendicular. 1 2 A = d1d2Area of a rhombus 1 2 A = (24)(10) Substitute 24 for d1 and 10 for d2. Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Additional Examples The diagonals of a rhombus bisect each other, so TX = 12 ft. You can use the Pythagorean triple 5, 12, 13 or the Pythagorean Theorem to conclude that SX = 5 ft. SU = 10 ft because the diagonals of a rhombus bisect each other. A = 120 Simplify. Quick Check The area of rhombus RSTU is 120 ft2. 10-2

15. 94.5 3 in.2 100 2 m2 Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Lesson Quiz 1. Find the area of a trapezoid with bases 3 cm and 19 cm and height 9 cm. 2. Find the area of a trapezoid in a coordinate plane with vertices at (1, 1), (1, 6), (5, 9), and (5, 1). Find the area of each figure in Exercises 3–5. Leave your answers in simplest radical form. 3. trapezoid ABCD 4. kite with diagonals 20 m and 10 2 m long 5. rhombus MNOP 99 cm2 26 square units 840 mm2 10-2