MATH 125 Section 3.4 - Pasadena City College | MATH125 Section 3.4 - Pasadena City College

MATH 125 Section 3.4 - Pasadena City College | MATH125 Section 3.4 - Pasadena City College Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem Learning Objectives By the end of this section, you will be able to: Solve applications using properties of triangles Use the Pythagorean Theorem Solve applications using rectangle properties Be Prepared! Before you get started, take this readiness quiz. 1. Simplify: 1 2(6h). If you missed this problem, review Example 1.122. 2. The length of a rectangle is three less than the width. Let w represent the width. Write an expression for the length of the rectangle. If you missed this problem, review Example 1.26. 3. Solve: A = 1 2bh for b when A = 260 and h = 52. If you missed this problem, review Example 2.61. 4. Simplify: 144. If you missed this problem, review Example 1.111. Solve Applications Using Properties of Triangles In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications. We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex. The plural of the word vertex is vertices. All triangles have three vertices. Triangles are named by their vertices: The triangle in Figure 3.4 is called △ABC. HOW TO : : SOLVE GEOMETRY APPLICATIONS. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information. Identify what we are looking for. Label what we are looking for by choosing a variable to represent it. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Solve the equation using good algebra techniques. Check the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem. Answer the question with a complete sentence. Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. 346 Chapter 3 Math Models This OpenStax book is available for free at and C. The lengths of the sides are a, b, and c. The three angles of a triangle are related in a special way. The sum of their measures is 180°. Note that we read m∠A as “the measure of angle A.” So in △ABC in Figure 3.4, m∠A + m∠B + m∠C = 180° Because the perimeter of a figure is the length of its boundary, the perimeter of △ABC is the sum of the lengths of its three sides. P = a + b + c To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a 90° angle with the base. We will draw △ABC again, and now show the height, h. See Figure 3.5. Figure 3.5 The formula for the area of △ABC is A = 1 2bh, where b is the base and h is the height. Triangle Properties For △ABC Angle measures: m∠A + m∠B + m∠C = 180 • The sum of the measures of the angles of a triangle is 180°. Perimeter: P = a + b + c Chapter 3 Math Models 347• The perimeter is the sum of the lengths of the sides of the triangle. Area: A = 1 2bh, b = base, h = height • The area of a triangle is one-half the base times the height. EXAMPLE 3.34 The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle. Solution Step 1. Read the problem. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. the measure of the third angle in a triangle Step 3. Name. Choose a variable to represent it. Let x = the measure of the angle. Step 4. Translate. Write the appropriate formula and substitute. m ∠ A + m ∠ B + m ∠ C = 180 Step 5. Solve the equation. 55 + 82 + x = 180 137 + x = 180 x = 43 Step 6. Check. 55 + 82 + 43 ≟ 180 180 = 180 Step 7. Answer the question. The measure of the third angle is 43 degrees. TRY IT : : 3.67 The measures of two angles of a triangle are 31 and 128 degrees. Find the measure of the third angle. TRY IT : : 3.68 The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the third angle. EXAMPLE 3.35 The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine feet. How long is the third side? 348 Chapter 3 Math Models This OpenStax book is available for free at Step 1. Read the problem. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. length of the third side of a triangle Step 3. Name. Choose a variable to represent it. Let c = the third side. Step 4. Translate. Write the appropriate formula and substitute. Substitute in the given information. Step 5. Solve the equation. Step 6. Check. P = a + b + c 24 ≟ 4 + 9 + 11 24 = 24 Step 7. Answer the question. The third side is 11 feet long. TRY IT : : 3.69 The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side? TRY IT : : 3.70 The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is 18 feet. How long is the third side? EXAMPLE 3.36 The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height? Chapter 3 Math Models 349Solution Step 1. Read the problem. Draw the figure and label it with the given information. Area = 90m2 Step 2. Identify what you are looking for. height of a triangle Step 3. Name. Choose a variable to represent it. Let h = the height. Step 4. Translate. Write the appropriate formula. Substitute in the given information. Step 5. Solve the equation. Step 6. Check. A = 1 2bh 90 ≟ 1 2 ⋅ 15 ⋅ 12 90 = 90 Step 7. Answer the question. The height of the triangle is 12 meters. TRY IT : : 3.71 The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height? TRY IT : : 3.72 A triangular tent door has area 15 square feet. The height is five feet. What is the base? The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one 90° angle, which we usually mark with a small square in the corner. Right Triangle A right triangle has one 90° angle, which is often marked with a square at the vertex. 350 Chapter 3 Math Models This OpenStax book is available for free at One angle of a right triangle measures 28°. What is the measure of the third angle? Solution Step 1. Read the problem. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. the measure of an angle Step 3. Name. Choose a variable to represent it. Let x = the measure of an angle. Step 4. Translate. m ∠ A + m ∠ B + m ∠ C = 180 - - - -- - -- - - - - - Continued