LESSON 8–1 Geometric Mean

Five-Minute Check (over Chapter 7) TEKS Then/Now New Vocabulary Key Concept: Geometric Mean Example 1: Geometric Mean Theorem 8.1 Example 2: Identify Similar Right Triangles Theorems: Right Triangle Geometric Mean Theorems Example 3: Use Geometric Mean with Right Triangles Example 4: Real-World Example: Indirect Measurement Lesson Menu

Solve the proportion A. 15 B. 16.5 C. D. 18 5-Minute Check 1

The triangles at the right are similar. Find x and y. A. x = 16.1, y = 1.6 B. x = 15.6, y = 2.1 C. x = 7.8, y = 8.4 D. x = 17.6, y = 3.7 5-Minute Check 2

D. No, sides are not proportional. Determine whether the triangles are similar. If so, write a similarity statement. A. yes, ΔABC ~ ΔEDF B. yes, ΔABC ~ ΔDEF C. yes, ΔABC ~ ΔEFD D. No, sides are not proportional. 5-Minute Check 3

Find the perimeter of ΔDEF if ΔABC ~ ΔDEF, AB = 6. 3, DE = 15 Find the perimeter of ΔDEF if ΔABC ~ ΔDEF, AB = 6.3, DE = 15.75, and the perimeter of ΔABC is 26.5. A. 66.25 B. 64.5 C. 12.4 D. 10.6 5-Minute Check 4

Figure JKLM has a perimeter of 56 units Figure JKLM has a perimeter of 56 units. After a dilation with a scale factor of 2, what will the perimeter of figure J'K'L'M' be? A. 28 units B. 54 units C. 58 units D. 112 units 5-Minute Check 5

Mathematical Processes G.1(A), G.1(E) Targeted TEKS G.8(B) Identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems. Mathematical Processes G.1(A), G.1(E) TEKS

Find the geometric mean between two numbers. You used proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles. Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. Then/Now

geometric mean Vocabulary

Concept

Find the geometric mean between 2 and 50. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10. Example 1

A. Find the geometric mean between 3 and 12. Example 1

Concept

Separate the triangles into two triangles along the altitude. Identify Similar Right Triangles Write a similarity statement identifying the three similar triangles in the figure. Separate the triangles into two triangles along the altitude. Example 2

Answer: So, by Theorem 8.1, ΔEGF ~ ΔFGH ~ ΔEFH. Identify Similar Right Triangles Then sketch the three triangles, reorienting the smaller ones so that their corresponding angles and sides are in the same position as the original triangle. Answer: So, by Theorem 8.1, ΔEGF ~ ΔFGH ~ ΔEFH. Example 2

Write a similarity statement identifying the three similar triangles in the figure. A. ΔLNM ~ ΔMLO ~ ΔNMO B. ΔNML ~ ΔLOM ~ ΔMNO C. ΔLMN ~ ΔLOM ~ ΔMON D. ΔLMN ~ ΔLMO ~ ΔMNO Example 2

Concept

Use Geometric Mean with Right Triangles Find c, d, and e. Example 3

Geometric Mean (Altitude) Theorem Use Geometric Mean with Right Triangles Since e is the measure of the altitude drawn to the hypotenuse of right ΔJKL, e is the geometric mean of the lengths of the two segments that make up the hypotenuse, JM and ML. Geometric Mean (Altitude) Theorem Substitution Simplify. Example 3

Geometric Mean (Leg) Theorem Use Geometric Mean with Right Triangles Since d is the measure of leg JK, d is the geometric mean of JM, the measure of the segment adjacent to this leg, and the measure of the hypotenuse JL. Geometric Mean (Leg) Theorem Substitution Use a calculator to simplify. Example 3

Geometric Mean (Leg) Theorem Use Geometric Mean with Right Triangles Since c is the measure of leg KL, c is the geometric mean of ML, the measure of the segment adjacent to KL, and the measure of the hypotenuse JL. Geometric Mean (Leg) Theorem Substitution Use a calculator to simplify. Answer: e = 12, d ≈ 13.4, c ≈ 26.8 Example 3

Find e to the nearest tenth. B. 24 C. 17.9 D. 11.3 Example 3

Indirect Measurement KITES Ms. Alspach is constructing a kite for her son. She has to arrange two support rods so that they are perpendicular. The shorter rod is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod? Example 4

Draw a diagram of one of the right triangles formed. Indirect Measurement Draw a diagram of one of the right triangles formed. Let be the altitude drawn from the right angle of ΔWYZ. Example 4

Geometric Mean (Altitude) Theorem Indirect Measurement Geometric Mean (Altitude) Theorem Substitution Square each side. Divide each side by 7.25. Answer: The length of the long rod is 7.25 + 25.14, or about 32.39 inches long. Example 4

AIRPLANES A jetliner has a wingspan, BD, of 211 feet AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, at point E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot? A. 68.3 ft B. 231.3 ft C. 273.1 ft D. 436.1 ft Example 4

LESSON 8–1 Geometric Mean