Lesson 7-6

Slide 1:

5-Minute Check on Lesson 7-5 Transparency 7-6 Click the mouse button or press the Space Bar to display the answers. Refer to the figure If QT = 5, TR = 4, and US = 6, find QU. If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x . Refer to the figure If AB = 5, ED = 8, BC = 11, and DC = x – 2, find x so that BD // AE. If AB = 4, BC = 7, ED = 5, and EC = 13.75, determine whether BD // AE. 5. Find the value of x + y in the figure? Standardized Test Practice: A C B D 4 6 8 10 7.5 Yes 19.6 B 3 R T Q U S A B C D E 5 y – 6 2 y + 3 3 x – 2 2 x + 1

Lesson 7-6:

Lesson 7-6 Parts of Similar Triangles ( Pages 342 – 348 in our text )

Objectives:

Objectives Recognize and use proportional relationships BETWEEN the perimeters AND the corresponding sides of SIMILAR triangles. Recognize and use proportional relationships BETWEEN corresponding angle bisectors , altitudes , and medians AND the corresponding sides of SIMILAR triangles

Vocabulary:

Vocabulary None New

Theorems:

Theorems If two triangles are similar then The perimeters are proportional to the measures of corresponding sides The measures of the corresponding altitudes are proportional to the measures of the corresponding sides The measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides The measures of the corresponding medians are proportional to the measures of the corresponding sides Theorem 7.11: Angle Bisector Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides

Special Segments of Similar Triangles:

Special Segments of Similar Triangles If ∆PMN ~ ∆PRQ, then PM PN MN special segment ----- = ----- = ----- = ------------------------- AB AC BC special segment ratios of corresponding special segments = Scale Factor (just like the sides) in similar triangles Example: PM 1 median PQ 1 ----- = --- ----------------- = --- AB 3 median AD 3 Special segments are altitudes, medians, angle and perpendicular bisectors P M A N B C D Q

Similar Triangles -- Perimeters:

Similar Triangles -- Perimeters P Q R M N If ∆PMN ~ ∆PRQ, then Perimeter of ∆PMN PM PN MN ------------------------- = ----- = ----- = ----- Perimeter of ∆PRQ PR PQ RQ ratios of perimeters = Scale Factor (just like the sides)

Angle Bisector Theorem - Ratios:

Angle Bisector Theorem - Ratios P Q R N If PN is an angle bisector of P, then the ratio of the divided opposite side, RQ, is the same as the ratio of the sides of P, PR and PQ PR RN ----- = ----- PQ NQ

Example 1a:

If ∆ ABC ~∆ XYZ , AC=32, AB=16, BC=16 5, and XY=24, find the perimeter of ∆ XYZ Let x represent the perimeter of The perimeter of C Example 1a

Example 1a cont:

Proportional Perimeter Theorem Substitution Cross products Multiply. Divide each side by 16. Answer: The perimeter of Example 1a cont

Example 1b:

If ∆PNO~∆XQR, PN=6, XQ=20, QR=20 2, and RX = 20 , find the perimeter of ∆PNO Answer: R Example 1b

Example 2a:

∆ ABC ~ ∆ MNO by a ratio of 1:3 (because from 3 BC = NO, we get BC = 1/3NO) or ∆ MNO ~ ∆ ABC by a ratio of 3:1 . According to Theorem 7.8, if two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. Answer: The ratio of the lengths of the altitudes is 1:3 or ⅓ ∆ ABC ~∆ MNO and 3BC = NO. Find the ratio of the length of an altitude of ∆ ABC to the length of an altitude of ∆ MNO Example 2a

Example 2b:

Answer: ∆EFG~∆MSY and 4EF = 5MS. Find the ratio of the length of a median of ∆EFG to the length of a median of ∆MSY. Example 2b

Example 3a:

In the figure, ∆ EFG~ ∆ JKL, ED is an ALTITUDE of ∆ EFG and JI is an ALTITUDE of ∆ JKL. Find x if EF=36, ED=18, and JK=56. K Write a proportion. Cross products Divide each side by 36. Answer: Thus, JI = 28. Example 3a

Example 3b:

Answer: 17.5 N In the figure, ∆ ABD ~ ∆ MNP and AC is a ANGLE BISECTOR of ∆ ABD and MO is a ANGLE BISECTOR of ∆ MNP. Find x if AC =5, AB =7 and MO =12.5 Example 3b

Example 4:

The drawing below illustrates two poles supported by wires with ∆ABC~∆GED , AF CF , and FG GC DC. Find the height of the pole EC. are ALTITUDES of since and If two triangles are similar, then the measures of the corresponding ALTITUDES are proportional to the measures of the corresponding sides . This leads to the proportion Example 4

Example 4 cont:

measures 40 ft. Also, since both measure 20 ft. Therefore, Write a proportion. Cross products Simplify. Divide each side by 80. Answer: The height of the pole is 15 feet. Example 4 cont

Slide 18:

5-Minute Check on Lesson 6-5 Transparency 6-6 Click the mouse button or press the Space Bar to display the answers. Find the perimeter of the given triangle. ∆UVW, if ∆UVW ~ ∆UVW, MN = 6, NP = 8, MP = 12, and UW = 15.6 ∆ABC, if ∆ABC ~ ∆DEF, BC = 4.5, EF = 9.9, and the perimeter of ∆DEF is 40.04. Find x. 3. 4. 5. Find NO, if ∆MNO ~ ∆RSQ. Standardized Test Practice: A C B D 3.67 6.75 7 8.25 33.8 x = 7.375 x = 6 D 18.2 2 x 9 8 x 8.5 12 x – 1 9 5.5 R Q S T 3 4.5 O P N M

Slide 19:

B C A K L J 3x + 1 5x - 1 8 12 A B C D F E x x - 2 6 12 12 16 Find x and the perimeter of DEF, if ∆DEF ~ ∆ABC Find x P S R T 8 x 6 x + 2 Find x if PT is an angle bisector A B C D E Find x, ED, and DB if ED = x – 3, CA = 20, EC = 16, and DB = x + 5 A B C D 6 4 P M N y 9 15 L Find y, if ∆ABC ~ ∆PNM Is CD a midsegment (connects two midpoints)?

Slide 20:

B C A K L J 3x + 1 5x - 1 8 12 A B C D F E x x - 2 6 12 12 16 Find x P S R T 8 x 6 x + 2 Find x if PT is an angle bisector A B C D E Find x, ED, and DB if ED = x – 3, CA = 20, EC = 16, and DB = x + 5 A B C D 6 4 P M N y 9 15 L Find y, if ∆ABC ~ ∆PNM 3x + 1 8 ------- = ---- 5x – 1 12 36x + 12 = 40x – 8 20 = 4x 5 = x Find x and the perimeter of DEF, if ∆DEF ~ ∆ABC x 6 --- = ---- 16 12 12x = 96 x = 8 P = (x – 2) + x + 6 = 2x + 4 = 2(8) + 4 = 20 x – 3 16 ------- = ---- x + 5 20 20x - 60 = 16x + 80 4x = 140 x = 35 ED = 32 DB = 40 8 6 ------- = ---- x + 2 x 8x = 6x + 12 2x = 12 x = 6 6 4 ------- = ---- 9 y 6y = 36 y = 6 Is CD a midsegment (connects two midpoints)? Since AC ≠ EC, then NO !

Summary & Homework:

Summary & Homework Summary: Similar triangles have perimeters proportional to the corresponding sides Corresponding angle bisectors, medians, and altitudes of similar triangles have lengths in the same ratio as corresponding sides WORK / Activity: (until revised for better teaching/learning) In-Class: Quiz on Previous Lesson, (I) Teach New Lesson, Recitation, Boardwork, Take notes… On-Line: Do Presentation on the new lesson, practice @least 3 times on the Quizzes, Take notes…