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Copyright © Cengage Learning. All rights reserved. Section 2-3 Models and Patterns in Right Triangles P.1 Velocity Copyright © Cengage Learning. All rights reserved.

Models and Patterns in Right Triangles In a right triangle, the sides of the triangle have lengths that demonstrate certain relationships. One of these relationships is stated by the Pythagorean theorem.

Models and Patterns in Right Triangles Figure 2-13

Models and Patterns in Right Triangles This theorem is only true for right triangles. It allows us to calculate the length of the third side of a right triangle if the lengths of the other two sides are known. Note that the hypotenuse (c) is always the longest side of the triangle and is directly opposite the right angle. The other two sides of the triangle (a and b) are called the legs.

Example 2 – Guy Wires for the Communications Company A 40-foot tower will be constructed for a cellular communications company. Guy wires will be needed for stability and will be attached 5 feet below the top of the tower. The wires will be anchored to the ground at a distance of 50 feet from the base of the tower. (Assume the tower makes a right angle with level ground.) What length of wire will be needed if the tower is to be stabilized with four guy wires? (See Figure 2-15.) Figure 2-15

Example 2 – Guy Wires for the Communications Company cont’d

Example 2 – Guy Wires for the Communications Company cont’d Because we need to have four guy wires, the total amount of wire needed will be: or approximately 245 ft to complete the job.

Models and Patterns in Right Triangles In trying to solve some geometry problems, figures can be subdivided into right triangles in order to calculate certain lengths. An isosceles triangle, for example, can be divided into two congruent right triangles by drawing in a perpendicular segment called the height from the vertex angle of the triangle to the base. (See Figure 2-16.) Figure 2-16

Example 4 – The Area of a Composite Figure Using the Pythagorean Theorem Find the painted area of the end of the house pictured in Figure 2-18. Do not use Hero’s formula to find this area. Figure 2-18

Example 4 – The Area of a Composite Figure Using the Pythagorean Theorem cont’d To find the area of the end of the house, we divide it into a rectangular portion and a triangle. We calculate the area of the rectangular portion using the formula A = lw, and then subtract the area of the two windows as calculated with the same formula.

Example 4 – The Area of a Composite Figure Using the Pythagorean Theorem cont’d Therefore, is the painted area of the rectangular portion of the house. Next, we need to calculate the area of the triangular portion of the end of the house. The easiest area formula for a triangle is A = 0.5bh. In this problem, the base is 24 ft. We do not know the height, but we can calculate the height using the Pythagorean theorem.

Area of the triangular portion = Example 4 – The Area of a Composite Figure Using the Pythagorean Theorem cont’d Therefore, Area of the triangular portion = Finally, to find the total painted area of the end of the house, we add our two results. of painted surface

The Standard or Conventional Method for Lettering a Triangle

The Standard or Conventional Method for Lettering a Triangle Look at the right triangle in Figure 2-19. Each angle is named using a capital letter A, B, and C. Each side is labeled with a lowercase letter corresponding to one of the letters used to name the angles. The standard method for labeling the sides of a triangle. Figure 2-19

The Standard or Conventional Method for Lettering a Triangle Notice that side a is the side opposite angle A, side b is opposite angle B, and side c is opposite angle C. The same is true for the triangle with angles labeled D, E, and F and sides labeled d, e, and f.