1. Copyright © Cengage Learning. All rights reserved. 12 Geometry

2. Copyright © Cengage Learning. All rights reserved. Triangles 12.3

3. 3 Triangles Triangles are often classified in two ways: 1. by the number of equal sides 2. by the measures of the angles of the triangle Triangles may be classified or named by the relative lengths of their sides.

4. 4 Triangles In each triangle in Figure 12.28, the lengths of the sides are represented by a, b, and c. An equilateral triangle has all three sides equal. All three angles are also equal. An isosceles triangle has two sides equal. The angles opposite these two sides are also equal. A scalene triangle has no sides equal. No angles are equal either. (a) Equilateral triangle (b) Isosceles triangle (c) Scalene triangle Figure 12.28 Triangles named by sides

5. 5 Triangles Triangles may also be classified or named in terms of the measures of their angles (see Figure 12.29). A right triangle has one right angle. An acute triangle has three acute angles. An obtuse triangle has one obtuse angle. (a) Right triangle(b) Acute triangle (c) Obtuse triangle Figure 12.29 Triangles named by angles

6. 6 Pythagorean Theorem

7. 7 In a right triangle, the side opposite the right angle is called the hypotenuse, which we label c. The other two sides, the sides opposite the acute angles, are called legs, which we label a and b. (See Figure 12.29a.) The Pythagorean theorem relates the lengths of the sides of any right triangle as follows: Right triangle Figure 12.29(a)

8. 8 Pythagorean Theorem The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the two legs. Alternative forms of the Pythagorean theorem are a 2 = c 2 – b 2 or and b 2 = c 2 – a 2 or

9. 9 Example 1 Find the length of the hypotenuse of the triangle in Figure 12.30. Substitute 5.00 cm for a and 12.0 cm for b in the formula: Figure 12.30

10. 10 Example 1 Note: You may need to use parentheses with some calculators. cont’d

11. 11 Perimeter and Area

12. 12 Perimeter and Area To find the perimeter of a triangle, find the sum of the lengths of the three sides. The formula is P = a + b + c, where P is the perimeter and a, b, and c are the lengths of the sides.

13. 13 Perimeter and Area An altitude of a triangle is a line segment drawn perpendicular from one vertex to the opposite side. Sometimes this opposite side must be extended. See Figure 12.34. Figure 12.34

14. 14 Perimeter and Area Look closely at a parallelogram (Figure 12.35a) to find the formula for the area of a triangle. Remember that the area of a parallelogram with sides of lengths a and b is given by A = bh. In this formula, b is the length of the base of the parallelogram, and h is the length of the altitude drawn to that base. Figure 12.35(a)

15. 15 Perimeter and Area Next, draw a line segment from B to D in the parallelogram as in Figure 12.35(b). Two triangles are formed. We know from geometry that these two triangles have equal areas. Since the area of the parallelogram is bh square units, the area of one triangle is one-half the area of the parallelogram. Figure 12.35(b)

16. 16 Perimeter and Area So the formula for the area of a triangle is

17. 17 Example 5 The length of the base of a triangle is 10.0 cm. The length of the altitude to that base is 6.00 cm. Find the area of the triangle. (See Figure 12.36.) A = bh A = (10.0 cm)(6.00 cm) = 30.0 cm 2 The area is 30.0 cm 2. Figure 12.36

18. 18 Perimeter and Area If only the lengths of the three sides are known, the area of a triangle is found by the following formula (called Heron’s formula):

19. 19 Example 7 Find the perimeter and the area (rounded to three significant digits) of the triangle in Figure 12.38. P = a + b + c P = 9 cm + 15 cm + 18 cm = 42 cm Figure 12.38

20. 20 Example 7 To find the area, first find s. s = (a + b + c) s = (9 + 15 + 18) = (42) = 21 cont’d

21. 21 Example 7 cont’d

22. 22 Example 7 cont’d

23. 23 Perimeter and Area The following relationship is often used in geometry and trigonometry. Figure 12.39