Date: Section 6 1: Proportions Notes Write Ratios: Ratio: Ways to express the ratio a to b: Example #1: The total number of students who participate in sports programs at Woodland Hills High School is 703. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. Extended Ratios in Triangles: Example #2: In a triangle, the ratio of the measures of three sides is 5:12:13, and the perimeter is 90 centimeters. Find the measure of the shortest side of the triangle. 1
Properties of Proportions: Proportion: Cross Products: Extremes: Means: Example #3: Solve each proportion. a.) 3 5 = x 75 b.) 3x 5 13 = 4 2 c.) 2.3 y = 4 3.4 Example #4: A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model. 2
IDENTIFY SIMILAR FIGURES Similar Polygons: Section 6 2: Similar Polygons Notes Date: Key Concept Two polygons are if and only if their corresponding are congruent and the measures of their corresponding sides are. Symbol: Similarity statement: Congruent angles: Corresponding sides: 1
Example #1: Determine whether the pair of figures is similar. Justify your answer. Scale Factor a numerical when comparing the lengths of corresponding of similar figures Example #2: Some special effects in movies are created using miniature models. In a recent movie, a model SUV 22 inches long was created to look like a real 14 2/3-foot SUV. What is the scale factor of the model compared to the real SUV? Example #3: The two polygons are similar. (a) Write a similarity statement. (b) Find x, y, and UV. (c) Find the scale factor of polygon ABCDE to polygon RSTUV. a.) b.) c.) 2
IDENTIFY SIMILAR TRIANGLES Section 6 3: Similar Triangles Notes Date: Angle-Angle (AA) Similarity: If the two of one triangle are to two angles of another triangle, then the triangles are. Side-Side-Side (SSS) Similarity: If the measures of the corresponding of two triangles are, then the triangles are similar. Side-Angle-Side (SAS) Similarity: If the measures of two of a triangle are proportional to the measures of two corresponding sides of another triangle and the included are congruent, then the triangles are. 1
Example #1: In the figure, FG EG, BE = 15, AE = 9, and DF = 12. Determine which triangles in the figure are similar. Example #2: Given RS TU, RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT. Example #3: Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 pm. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower s shadow and it was 242 feet at that time. What is the height of the Sears Tower? 2
Date: Section 6 4: Parallel Lines and Proportional Parts Notes PROPORTIONAL PARTS OF TRIANGLES Triangle Proportionality Theorem: If a line is to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of lengths. Example #1: In RST, RT VU, SV = 3, VR = 8, and UT = 12. Find SU. Converse of the Triangle Proportionality Theorem: If a line intersects two sides of a and separates the sides into corresponding segments of proportional, then the line is to the third side. 1
Example #2: In DEF, DH = 18, HE = 36, and DG = ½ GF. Determine whether GH FE. Explain! Triangle Midsegment Theorem: A midsegment of a triangle is to one side of the triangle, and its length is the length of that side. Example #3: In the figure, OA is a midsegment of MTH. Find x and y. 2
Section 6 5: Parts of Similar Triangles Notes Date: PERIMETERS Perimeter: Theorem 6.7: Proportional Perimeters Theorem If two triangles are similar, then the are proportional to the measures of the sides. If LMN ~ QRS, QR = 35, RS = 37, SQ = 12, and NL = 5, find the perimeter of LMN. Theorem 6.8: If two triangles are similar, then the of the corresponding are proportional to the measures of the corresponding sides. 1
Theorem 6.9: If two triangles are similar, then the measures of the corresponding are proportional to the measures of the corresponding sides. Theorem 6.10: If two triangles are similar, then the measures of the corresponding are proportional to the measures of the corresponding sides. Example #1: Draw ABC ~ DEF. BG is a median of ABC, and EH is a median of DEF. Find EH if BC = 30, BG = 15, and EF = 15. 2