Lesson 11.4 • Corresponding Parts of Similar Triangles

Lesson 11.4 • Corresponding Parts of Similar Triangles (continued)Step 7To finish constructing triangle ABC, hide all partsof triangle ABC except point A and construct themissing segments of the new triangle. Change yourratio and observe the results.BBIn Steps 8–10, you’ll construct altitudes in the similartriangles and measure their ratio.AmnACm__n 0.7CStep 8Select a vertex of one triangle and the side oppositethat vertex, and choose Construct ⏐ Perpendicular Line.Construct the point where the perpendicular intersectsthe side, construct the segment, and hide the line.Step 7Step 9Step 10Do the same in the other triangle.To measure the ratio between the altitudes,select the shorter altitude, then the longeraltitude, and choose Measure ⏐ Ratio.ABBC ACorresponding AltitudesCInvestigate1. Drag parts of your sketch and observe the ratio of the altitudes. Recordyour observations.SketchIn Steps 11 and 12, you’ll construct medians in your triangles and measuretheir ratio.Step 11Step 12Delete or hide the altitudes, and construct a pair of correspondingmedians.Measure the ratio of the medians. (For consistency, select themedian on ABC first.)BBACACCorresponding MediansInvestigate2. Drag parts of your sketch and observe the ratio of the medians. Recordyour observations.(continued)142 CHAPTER 11 Discovering Geometry with The Geometer’s Sketchpad©2008 Key Curriculum Press

Lesson 11.4 • Corresponding Parts of Similar Triangles (continued)SketchIn Steps 13 and 14, you’ll construct angle bisectors and measure their ratio.Step 13Step 14Delete or hide the medians, and construct a pair of correspondingangle bisectors. (To construct an angle bisector, select three pointson the angle, with the vertex in the middle. Choose Construct ⏐Angle Bisector. This gives you a ray. Construct the point wherethe ray intersects the opposite side, construct a segment, and hidethe ray.)Measure the ratio of the angle bisectors. (For consistency, select theshorter one first.)BBACACCorresponding Angle BisectorsInvestigate3. Drag parts of your sketch and observe the ratio of the angle bisectors.Record your observations.4. Combine your observations about altitudes, medians, and anglebisectors into a single conjecture (Proportional Parts Conjecture).Investigation 2: Opposite Side RatiosSketchStep 1In a new sketch, use the Ray tool to constructan angle BAC.ABStep 2Bisect the angle by selecting points B, A, and C inthat order, then choosing Construct ⏐ Angle Bisector.CDStep 3Step 4Step 5Construct a point D on the angle bisector and usethe Line tool to draw a line through segment DB.Construct the intersection E of that line and theother side of the angle.Construct segments ED, DB, BA, and AE. Hide the line and the raysof the original angle.Select all segments and measure their lengths.EInvestigate1. Drag point B and look for patterns. Try calculating some ratios ofsegment lengths. Write the Angle Bisector/Opposite Side Conjectureto summarize your findings.Discovering Geometry with The Geometer’s Sketchpad CHAPTER 11 143©2008 Key Curriculum Press