7.5 Proportions and. Similar Triangles. Geo-Activity. Goal Use the Triangle Proportionality Theorem and its converse.
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1 7. roportions and imilar riangles Goal Use the riangle roportionalit heorem and its converse. Ke Words midsegment of a triangle Geo-ctivit 1 raw a triangle. Label its vertices,, and. Make sure that each side is at least cm. raw a point on &*. Label the point. Investigating roportional egments 2 raw a line through parallel to &*. Label the intersection of the line and &* as point. 3 Measure &*, &*, &*, and &* in centimeters. hen calculate the ratios and. Make a conjecture about the ratios of segment lengths of a triangle s sides when the triangle is cut b a line parallel to the triangle s third side. roportionalit uppose that a point lies on GH &** and a point Q lies on G JQ JK&*. If, then we sa that GH &* and JK&* are divided proportionall. H Q K G 3 6 H J K In the Geo-ctivit above, &* divides &* and &* proportionall. HOM 7. riangle roportionalit heorem Words If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionall. U U mbols In Q, if U&* Q &*, then. Q U 36 hapter 7 imilarit
2 XML 1 Find egment Lengths Find the value of. olution riangle roportionalit heorem ubstitute for, for, for, and for. p p ross product propert Multipl. ivide each side b. 6 implif. Itudent Help I LZON.OM MO XML More eamples at classzone.com XML 2 Find egment Lengths Find the value of olution You know that 20 and. the egment ddition ostulate, 20. Q Q riangle roportionalit heorem ubstitute 3 for Q, 9 for Q, for, and (20 ) for. 3(20 ) 9 p ross product propert istributive propert dd 3 to each side. 60 implif ivide each side b. implif. 7. roportions and imilar riangles 37
3 HOM 7. onverse of the riangle roportionalit heorem Words If a line divides two sides of a triangle proportionall, then it is parallel to the third side. mbols In Q, if U, then U&* Q &*. Q U U XML 3 etermine arallels Given the diagram, determine whether MN is parallel to GH. M 21 G 6 L N H olution Find and simplif the ratios of the two sides divided b MN. LM 6 M G 21 3 LN NH 3 1 NW ecause 3 3, MN is not parallel to GH 1. Find egment Lengths and etermine arallels Find the value of the variable Given the diagram, determine whether Q &* is parallel to &*. plain hapter 7 imilarit
4 midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. he following theorem about midsegments is a special case of the riangle roportionalit heorem. HOM 7.6 he Midsegment heorem Words he segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. mbols In, if and, then &* &* and 1 2. XML Use the Midsegment heorem Find the length of Q &*. olution From the marks on the diagram, ou know is the midpoint of &*, and Q is the midpoint of &*. herefore, Q &* is a midsegment of. Use the Midsegment heorem to write the following equation. Q () 2 NW he length of Q &* is. Use the Midsegment heorem Find the value of the variable.. 6. p q Use the Midsegment heorem to find the perimeter of roportions and imilar riangles 39
5 7. ercises Guided ractice Vocabular heck omplete the statement. 1. he? heorem states that if a line divides two sides of a triangle proportionall, then it is? to the third side. 2.? of a triangle is a segment that connects the midpoints of two sides of a triangle. kill heck op and complete the proportion using the diagram below.? 3..?.? 6.? Find the value of the variable z ractice and pplications tra ractice ee p. 6. Using lgebra olve the proportion m t r Finding egment Lengths Find the value of the variable p 6 20 Homework H lp ample 1: s ample 2: s ample 3: s ample : s. 2 29, q c z hapter 7 imilarit
6 etermining arallels Given the diagram, determine whether Q &* is parallel to &* Using the Midsegment heorem Find the value of the variable a b Fractals c z Visualize It! he design below approimates a fractal. egin with an equilateral triangle. hade the triangle formed b the three midsegments. ontinue the process for each unshaded triangle. FL are shapes that look the same at man levels of magnification. ake a small part of the image above and ou will see that it looks similar to the whole image. pplication Links LZON.OM tage 0 tage 1 tage 2 tage Find the perimeter of the dark blue triangle in tage hallenge Find the total perimeter of all the dark blue triangles in tage hallenge Find the total perimeter of all the dark blue triangles in tage roportions and imilar riangles 391
7 Midsegment heorem Use the diagram below to complete the statement. 33. LM? 3. &*? 3. If, then LN?. 36. If MN 7., then?. L N 37. If N 9., then LM?. M echnolog In ercises 3 and 39, use geometr software to complete the steps below. 1 raw. 2 onstruct the angle bisector of a. 3 onstruct the intersection of the angle bisector and &*. Label it. Measure & *, &*, &*, and &*. hen calculate the ratios and. 3. rag one or more of the triangle s vertices. What do ou notice about the ratios as the shape changes? 39. omplete the conjecture: If a ra bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are? to the lengths of the other two sides. tandardized est ractice 0. Multiple hoice What is the value of? Mied eview eflections etermine if the entire word has an lines of smmetr. If so, write the word and draw the line(s) of smmetr. (Lesson.7) lgebra kills Finding lope Find the slope of the line that passes through the points. (kills eview, p. 66). (0, 2) and (, ). (1, 2) and (3, ) 6. (, 2) and (, 3) 7. (, 6) and (1, 2). (, ) and (2, 0) 9. (3, 7) and (1, 3) 0. (, 3) and (1, 1) 1. (0, ) and (3, ) 2. (3, 2) and (6, ) 392 hapter 7 imilarit
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