Area and Perimeter of Rhombuses and Kites Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Bill Zahner Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). 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Printed: November 22, 2015 AUTHORS Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Bill Zahner www.ck12.org C HAPTER Chapter 1. Area and Perimeter of Rhombuses and Kites 1 Area and Perimeter of Rhombuses and Kites Here you’ll learn how to find the area and perimeter of a kite or a rhombus given its two diagonals. What if you were given a kite or a rhombus and the size of its two diagonals? How could you find the total distance around the kite or rhombus and the amount of space it takes up? After completing this Concept, you’ll be able to use the formulas for the perimeter and area of a kite/rhombus to solve problems like this. Watch This MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/136632 Area and Perimeter of Rhombuses and Kites CK-12 Guidance Recall that a rhombus is a quadrilateral with four congruent sides and a kite is a quadrilateral with distinct adjacent congruent sides. Both of these quadrilaterals have perpendicular diagonals, which is how we are going to find their areas. Notice that the diagonals divide each quadrilateral into 4 triangles. If we move the two triangles on the bottom of each quadrilateral so that they match up with the triangles above the horizontal diagonal, we would have two rectangles. So, the height of these rectangles is half of one of the diagonals and the base is the length of the other diagonal. 1 www.ck12.org The area of a rhombus or a kite is A = 21 d1 d2 Example A Find the perimeter and area of the rhombus below. In a rhombus, all four triangles created by the diagonals are congruent. To find the perimeter, you must find the length of each side, which would be the hypotenuse of one of the four triangles. Use the Pythagorean Theorem. 122 + 82 = side2 144 + 64 = side2 √ √ side = 208 = 4 13 √ √ P = 4 4 13 = 16 13 units Example B Find the perimeter and area of the rhombus below. 2 1 · 16 · 24 2 A = 192 units2 A= www.ck12.org Chapter 1. Area and Perimeter of Rhombuses and Kites In a rhombus, all four triangles created by the diagonals are congruent. Here, each triangle is a 30-60-90 triangle with a hypotenuse of 14. From the special right triangle ratios the short leg √ is 7 and the long leg is 7 3. P = 4 · 14 = 56 units A= √ √ 1 · 14 · 14 3 = 98 3 units2 2 Example C The vertices of a quadrilateral are A(2, 8), B(7, 9),C(11, 2), and D(3, 3). Show ABCD is a kite and find its area. After plotting the points, it looks like a kite. AB = AD and BC = DC. The diagonals are perpendicular if the slopes are negative reciprocals of each other. 2−8 6 2 =− =− 11 − 2 9 3 9−3 6 3 mBD = = = 7−3 4 2 mAC = The diagonals are perpendicular, so ABCD is a kite. To find the area, we need to find the length of the diagonals, AC and BD. q (2 − 11)2 + (8 − 2)2 q = (−9)2 + 62 √ √ √ = 81 + 36 = 117 = 3 13 q (7 − 3)2 + (9 − 3)2 p = 42 + 62 √ √ √ = 16 + 36 = 52 = 2 13 d1 = Plug these lengths into the area formula for a kite. A = d2 = 1 2 √ √ 3 13 2 13 = 39 units2 3 www.ck12.org MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/136633 Area and Perimeter of Rhombuses and Kites CK-12 –> Guided Practice Find the perimeter and area of the kites below. 1. 2. 3. Find the area of a rhombus with diagonals of 6 in and 8 in. Answers: In a kite, there are two pairs of congruent triangles. Use the Pythagorean Theorem in the first two problems to find the lengths of sides or diagonals. 1. Shorter sides of kite P=2 4 Longer sides of kite 62 + 52 = s21 122 + 52 = s22 36 + 25 = s21 √ s1 = 61 units 144 + 25 = s22 √ s2 = 169 = 13 units √ √ 61 + 2(13) = 2 61 + 26 ≈ 41.6 units 1 A = (10)(18) = 90 units 2 www.ck12.org Chapter 1. Area and Perimeter of Rhombuses and Kites 2. Smaller diagonal portion 2 20 + ds2 ds2 2 202 + dl2 = 352 = 25 dl2 = 825 √ dl = 5 33 units = 225 ds = 15 units A= Larger diagonal portion √ 1 15 + 5 33 (40) ≈ 874.5 units2 2 P = 2(25) + 2(35) = 120 units 3. The area is 12 (8)(6) = 24 in2 . Explore More 1. Do you think all rhombi and kites with the same diagonal lengths have the same area? Explain your answer. Find the area of the following shapes. Round your answers to the nearest hundredth. 2. 3. 4. 5 www.ck12.org 5. 6. 7. Find the area and perimeter of the following shapes. Round your answers to the nearest hundredth. 8. 9. 10. 6 www.ck12.org Chapter 1. Area and Perimeter of Rhombuses and Kites 11. For Questions 12 and 13, the area of a rhombus is 32 units2 . 12. What would the product of the diagonals have to be for the area to be 32 units2 ? 13. List two possibilities for the length of the diagonals, based on your answer from #12. For Questions 14 and 15, the area of a kite is 54 units2 . 14. What would the product of the diagonals have to be for the area to be 54 units2 ? 15. List two possibilities for the length of the diagonals, based on your answer from #14. Sherry designed the logo for a new company, made up of 3 congruent kites. 16. What are the lengths of the diagonals for one kite? 17. Find the area of one kite. 18. Find the area of the entire logo. Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 10.6. 7

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