Lesson 21: Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles

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1 : Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles Learning Targets I can state that the altitude of a right triangle from the vertex of the right angle to the hypotenuse divides the triangle into two similar right triangles that are also similar to the original right triangle. I can complete a table of ratios for the corresponding sides of the similar triangles that are the result of dividing a right triangle into two similar sub-triangles. Opening Exercise (5 minutes) Use the diagram below to complete parts (a) (c). a) Are the triangles shown above similar? b) Determine the unknown lengths of the triangles. c) Explain how you found the lengths in part (a). Definition: An of a triangle is a from a to the line determined by the side. In triangle ABC below, BD is the altitude from vertex B to the line containing AC. How many triangles do you see in the figure? Identify all triangles by name. Original Triangle: Sub- Triangles Is ABC ~ BDC? Explain. Is ABC ~ ADB? Explain. Is ABC~ DBC? Explain. Since ABC ~ BDC and ABC~ ADB, can we conclude that BDC~ ADB by the property

2 We can say that the altitude drawn from the right angle of a right triangle divides the triangle into two similar sub-triangles, which are also similar to the original triangle This fact allows us to determine the unknown lengths of right triangles. Example 2 (15 minutes) Consider the right triangle ABC below at right. Draw the altitude BD from vertex B to the line containing AC. Label the segment AD as x, the segment DC as y, and the segment BD as z. Fill in the table below with ratios of the side lengths of the triangle above: shorter leg: hypotenuse longer leg: hypotenuse shorter leg: longer leg ABC ADB CDB Our work in Example 1 showed us that ABC~ ADB~ CDB. Since the triangles are similar, the ratios of their corresponding sides will be equal. That means, you can choose any two ratios in the same column to create a proportion. a) Pick a pair of ratios from any column that you can use to make a solvable proportion. Solve for the variable. b) Pick another pair of ratios and solve another proportion. c) Solve for the last variable

3 short leg: hypot. long leg: hypot. short leg: long leg ABC AB : x + y CB : x + y AB : CB ADB x : AB z : AB x : z CDB z : CB y : CB z : y Choosing wisely leads us to the following 3 relationships that can help us solve for any missing segment: In words When the altitude is drawn to the hypotenuse in a right triangle: ~The length of the altitude squared is equal to the product of the lengths of the 2 segments of the hypotenuse that it creates ~ The length of either leg of the triangle squared is equal to the product of the lengths of the adjacent segment of the hypotenuse and the whole hypotenuse. Practice: Find the missing values. (If not a whole number, leave it in simplest radical form) a) b) c) x x z y z y x = y = z = x = y = z = x = y = z =

4 : Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles Classwork Exercise 1. Given RST, with altitude SU drawn to its hypotenuse, ST = 15, RS = 36, and RT = 39, answer the questions below. a) Complete the similarity statement relating the three triangles in the diagram. RST~ ~ b) Complete the table of ratios specified below. RST RSU STU shorter leg: hypotenuse longer leg: hypotenuse shorter leg: longer leg c) Use the values of the ratios you calculated to find the length of SU.

5 Exercise 2. Use similar triangles to find the length of the altitudes labeled with variables in each triangle below. a) b) x = y = c) z = Exercise 3. Given right triangle EFG with altitude FH drawn to the hypotenuse, find the lengths of EH, FH, and GH.

6 Exercise 4) Given triangle IMJ with altitude JL, JL = 32, and IL = 24, find IJ, JM, LM, and IM. IJ = JM = LM = IM = Exercise 5) In right triangle ABD, AB = 53, and altitude DC = 14. Find the lengths of BC and AC.

7 : Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles Homework Exercise 1. Given right triangle ABC with altitude CD, find AD, BD, AB, and DC. AD = AB = BD = DC = Using the Pythagorean theorem find the length of AB: AB = An altitude from the right angle in a right triangle to the hypotenuse cuts the triangle into two similar right triangles: Identify the three triangles using a similarity statement relating them: Using the appropriate ratios, find DC and DB. DC = DB = Using the Pythagorean Theorem find AD. AD = 2. In the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse AB. Which proportion would always represent a correct relationship of the segments? 1. c = z z y 2. c = a a y 3. x = z z y 4. y = b b x 3. In the diagram below of right triangle ABC, altitude CD is drawn to hypotenuse AB. DB = 12, what is the length of altitude CD? If AD = 3 and 1) 6 2) 6 5 3) 3 4) 3 5

8 4. The accompanying diagram shows a 24-foot ladder leaning against a building. A steel brace extends from the ladder to the point where the building meets the ground. The brace forms a right angle with the ladder. If the steel brace is connected to the ladder at a point that is 10 feet from the foot of the ladder, which equation can be used to find the length, x, of the steel brace? 1) 2) 3) 4) 5. The accompanying diagram shows part of the architectural plans for a structural support of a building. PLAN is a rectangle and AS LN. Which equation can be used to find the length of AS? 1) 2) 3) 4) 6. In the diagram below of right triangle ABC, altitude is drawn to hypotenuse,, and. What is the length of? 1) 2) 3) 4) 12