The Height of the triangle is the perpendicular segment from a vertex to the line containing the opposite side. The opposite side is called the Base of the triangle. The terms height and base are also used to represent the segment lengths.
1. Page 1 of 7 8.4 Area of Triangles Goal The amount of material needed to Find the area of triangles. make the sail at the right is determined by the area of the Key Words triangular sail. • height of a triangle The height and base of a triangle are • base of a triangle used to find its area. The height of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. The opposite side is called the base of the triangle . The terms height and base are also used to represent the segment lengths. height height height base base base In a right triangle, A height can be A height can be a leg is a height. inside the triangle. outside the triangle. As shown in Activity 8.4, the area of a triangle is found using a base and its corresponding height. AREA OF A TRIANGLE 1 Words Area 5 }} (base)(height) 2 1 Symbols A 5 }}bh 2 b Triangles with the Same Area Triangles can have the same area without necessarily being congruent. For example, all of the triangles below have the same area but they are not congruent. 8 8 8 8 13 13 13 13 8.4 Area of Triangles 431
3. Page 3 of 7 Area of Triangles In Exercises 1–3, find the area of the triangle. 1. 2. 3. 8 in. 7 yd 16 cm 15 cm 9 in. 12 yd 4. A triangle has an area of 84 square inches and a height of 14 inches. Find the base. EXAMPLE 4 Areas of Similar Triangles a. Find the ratio of the areas of the similar B triangles. 2 E b. Find the scale factor of T ABC to TDEF C 4 A and compare it to the ratio of their areas. 3 F 6 D Solution T ABC S TDEF 1 1 a. Area of T ABC 5 }}bh 5 }}(4)(2) 5 4 square units 2 2 1 1 Area of TDEF 5 }}bh 5 }}(6)(3) 5 9 square units 2 2 Student Help Area of T ABC 4 LOOK BACK Ratio of areas 5 }} 5 }} Area of TDEF 9 To review scale factor, 2 see p. 366. b. The scale factor of T ABC to TDEF is }}. 3 22 4 The ratio of the areas is the square of the scale factor: }}2 5 }}. 3 9 The relationship in Example 4 is generalized for all similar polygons in the following theorem. THEOREM 8.3 Areas of Similar Polygons Words If two polygons are similar with a scale C a G factor of }}, then the ratio of their areas B b a a2 is }}2 . F b b A D Symbols If ABCD S EFGH with a scale factor a Area of ABCD a2 of }}, then }} 5 }}2 . E H b Area of EFGH b 8.4 Area of Triangles 433
4. Page 4 of 7 8.4 Exercises Guided Practice Vocabulary Check 1. What are the measures of the base and 13 ft the height of the shaded triangle at the 5 ft right? 3 ft 9 ft Skill Check The triangle has a horizontal base of 15 units and a height of 7 units. Sketch the triangle and label its base and its height. 2. 3. 4. Practice and Applications Extra Practice Area of a Right Triangle Find the area of the right triangle. See p. 690. 5. 6. 7. 7 cm 6 ft 3 yd 5 yd 12 cm 7 ft Finding Area In Exercises 8–13, find the area of the triangle. 8. 9. 7 yd 10. 4m 8 mm 4 yd 9m 14 mm 11. 12. 13. 12 in. 9 ft 14 ft 4 cm 12 in. 5 cm 14. You be the Judge In the triangle Homework Help at the right, Trisha says the base is 15 and the height is 4. Luis says that the Example 1: Exs. 5–7 base is 5 and the height is 12. Who is 15 12 Example 2: Exs. 8–13 right? Explain your reasoning. Example 3: Exs. 16–20 4 Example 4: Exs. 26, 27 5 434 Chapter 8 Polygons and Area
5. Page 5 of 7 15. Visualize It! Draw three different triangles that each have an area of 24 square units. Using Algebra In Exercises 16–18, A gives the area of the triangle. Find the missing measure. 16. A 5 22 ft2 17. A 5 63 cm2 18. A 5 80 m2 b 4 ft h b 14 cm 16 m 19. Finding the Height A triangle has an area of 78 square inches and a base of 13 inches. Find the height. 20. Finding the Base A triangle has an area of 135 square meters and a height of 9 meters. Find the base. Tiles In Exercises 21 and 22, use the diagram of the tile pattern. 2 in. 4 in. 21. Find the area of one triangular tile. 22. The tiles are being used to make a rectangular border that is 4 inches high and 48 inches long. How many tiles are needed for the border? (Hint: Start by finding the area of the border.) Complex Polygons Find the area of the polygon by using the triangles and rectangles shown. 23. 24. 25. 8 cm 5 ft 4m 12 ft 6 cm 7 ft 4m 5 ft 5 cm 7m Areas of Similar Triangles In Exercises 26 and 27, the triangles are similar. Find the scale factor of TPQR to TXYZ. Then find the ratio of their areas. 26. P 27. P 3 Y Y 6 P 5 R 6 P 9 R 8 X 10 Z X 12 Z 8.4 Area of Triangles 435
6. Page 6 of 7 Area of a Regular Octagon In Exercises 28–30, use the regular octagon at the right. Geology B C 28. Find the area of TGXF in the octagon. A D 29. Copy the diagram. To form congruent X triangles, connect the following pairs of vertices: A and E, B and F, C and G, D H E and H. How many triangles are formed? G 4 F 30. What is the area of the octagon? Explain. 31. Rock Formations Many basaltic columns are hexagonal. The top of one of these columns is a regular hexagon as shown below. Find its area. (Another photograph of basaltic columns is on page 408.) ROCK FORMATIONS Geologists learn about the structure of the earth by studying rock formations such 7.8 in. as the basaltic columns at the Giant’s Causeway in Ireland pictured above. 9 in. Application Links CLASSZONE.COM EXAMPLE Using the Pythagorean Theorem Find the area of the triangle. 13 5 b Solution First, find the base. Use the Pythagorean Theorem to find the value of b. Student Help (hypotenuse)2 5 (leg)2 1 (leg)2 Pythagorean Theorem 2 2 2 LOOK BACK (13) 5 (5) 1 (b) Substitute. To review the 2 Pythagorean Theorem, 169 5 25 1 b Simplify. see pp. 192 and 193. 144 5 b2 Subtract 25 from each side. 12 5 b Find the positive square root. Use 12 as the base in the formula for the area of a triangle. 1 1 A 5 }}bh 5 }}(12)(5) 5 30 square units 2 2 Using the Pythagorean Theorem Find the area of the triangle. 32. 33. 34. 26 a 20 12 6 10 24 b b 436 Chapter 8 Polygons and Area
7. Page 7 of 7 Standardized Test 35. Multiple Choice Given that the area of the triangle is 99 square Practice meters, what is the height of the triangle? A 5 99 m2 X A 4.5 m B 9m X h X C 11 m D 22 m X 22 m Mixed Review Trapezoids Find the value of x in the trapezoid. (Lesson 6.5) 36. 6 37. 20 38. 18 x 16 21 x x 12 Algebra Skills Naming Coordinates Give the coordinates of the point. (Skills Review, p. 664) A y 39. A 40. B C 41. C 42. D B 1 43. E 44. F 1 x D E F Quiz 2 Find the area of the polygon. (Lessons 8.3, 8.4) 1. 2. 3. 5m 3 in. 7 in. 8m 12 cm 3m 7m 4. 5. 14 mm 6. 12 yd 14 ft 8 mm 16 yd 22 ft In Exercises 7–9, A gives the area of the polygon. Find the missing measure. (Lessons 8.3, 8.4) 7. A 5 48 in.2 8. A 5 90 m2 9. A 5 63 cm2 9 cm 6 in. h b 15 m b 8.4 Area of Triangles 437
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