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9-5 Proportions in Triangles Do Now Lesson Presentation Exit Ticket 9-5 Proportions in Triangles Warm Up #9 For each triangle, find the value of x. 1. x=6 3. x = 10 2. x = 20 9-5 Proportions in Triangles Find the value of x. 10 10 + π₯ π₯ + 14 = 10 π₯+2 10 + π₯ π₯ + 2 = 10 π₯ + 14 x+2 12π₯ + 20 + π₯ 2 = 10π₯ + 140 π₯ 2 + 2π₯ β 120 = 0 π₯ + 12 π₯ β 10 = 0 π₯ = β12 π₯ = 10 9-5 Proportions in Triangles 9-5 Proportions in Triangles Connect to Mathematical Ideas (1)(F) By the end of todayβs lesson, SWBAT ο§ Use the Triangle Proportionality Theorem and the Triangle-Angle-Bisector Theorem. ο§ Use geometric mean to find segment lengths in right triangles. ο§ Apply similarity relationships in right triangles to solve problems. 9-5 Proportions in Triangles Prior Knowledge: Find the value of x. 12 9 = π₯ + 13 π₯+9 12 π₯ + 9 = 9 π₯ + 13 12π₯ + 108 = 9π₯ + 117 3π₯ = 9 π₯=3 9-5 Proportions in Triangles 9-5 Proportions in Triangles Example 1: Find the value of x. ππΎ ππΏ β Proportionality Theorem. = πΎπ πΏπ π₯+1 π₯ = 12 9 Substitution 9 π₯ + 1 = 12π₯ Cross Products Property 9π₯ + 9 = 12π₯ Distributive Property 9 = 3π₯ Subtract 9x from each side π₯=3 Divide 3 from each side 9-5 Proportions in Triangles 9-5 Proportions in Triangles Example 2: Find the value of x. 9-5 Proportions in Triangles In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles. 9-5 Proportions in Triangles 9-5 Proportions in Triangles Example 3: Identifying Similar Triangles. What similarity statement can you write relating the three triangles in the diagram? ππΎ is the altitude to the hypotenuse of right βXYZ, so you can use Theorem 9-3. There are three similar triangles. 9-5 Proportions in Triangles π π₯ = . In this case, the Consider the proportion π₯ π means of the proportion are the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that π₯= 2 ππ, or π₯ = ππ. 9-5 Proportions in Triangles 9-5 Proportions in Triangles Example 4: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 4 and 25 Let x be the geometric mean. x2 = (4)(25) = 100 Def. of geometric mean x = 10 Find the positive square root. 9-5 Proportions in Triangles Example 5: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 5 and 30 Let x be the geometric mean. x2 = (5)(30) = 150 π₯ = 150 π₯ =5 6 Def. of geometric mean Find the positive square root. 9-5 Proportions in Triangles You can use Theorem 9-3 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means. ππ = ππ ππ = ππ ππ = ππ 9-5 Proportions in Triangles 9-5 Proportions in Triangles 9-5 Proportions in Triangles Example 6: Finding Side Lengths in Right Triangles Find x, y, and z. 62 = (9)(x) x=4 6 is the geometric mean of 9 and x. Divide both sides by 9. y2 = (4)(13) = 52 y is the geometric mean of 4 and 13. Find the positive square root. z2 = (9)(13) = 117 z is the geometric mean of 9 and 13. Find the positive square root. 9-5 Proportions in Triangles Got It ? Solve With Your Partner Problem 1 Finding the value of x. a. b. x = 7.5 c. x = 3.6 x = 35 9-5 Proportions in Triangles Got It ? Solve With Your Partner Problem 2 How far does the robot travel from A to D ? 9-5 Proportions in Triangles Got It ? Solve With Your Partner Problem 2 How far does the robot travel from A to D ? 9-5 Proportions in Triangles Closure: Communicate Mathematical Ideas (1)(G) ο§How is the geometric mean used in right triangles?. The altitude of a right triangle to the hypotenuse is the geometric mean of the segments of the hypotenuse it creates. A leg of a right triangle is the geometric mean of the hypotenuse and the segments of the hypotenuse created by the altitude, adjacent to the leg. ο§When parallel lines intersect two or more segments, what is the relationship between the segments formed? The segments formed between the parallel lines are proportional. 9-5 Proportions in Triangles Exit Ticket: Apply Mathematics (1)(A) A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot? Let x be the height of cliff above eye level. (28)2 = 5.5x 28 is the geometric mean of 5.5 and x. x ο» 142.5 Divide both sides by 5.5. The cliff is about 142.5 + 5.5, or 148 ft high.