Proportions & Similar Triangles

1. Proportions & Similar Triangles Investigation – Proportional parts of triangles Step 1 Use a ruler to draw a pair of parallel segments Step 2 Draw ∆ABC with A and C on one segment and B on the far side of the other segment Step 3 Label the points where AB and BCcross the second segment D and E Step 4 Measure BD, DA, BE, and EC Step 5 Compare the ratios between segments on the same sides of the triangle to each other (BD:DA and BE:EC) Step 6 Make a conjecture about the relationship between the ratio of the lengths of the parts of the two sides of a triangle cut by a line parallel to the third side Triangle Proportionality Theorem • If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally C E B A D

2. Proportions & Similar Triangles • In the figure at right, if DE || AC, then by the Triangle Proportionality Theorem = • The proof of the theorem uses both geometry and algebra Statement Reason DE || AC Given ÐA ÐBDE Corresponding angles are congruent ÐB ÐB Reflexive property of congruence ∆ABC ~ ∆DBE AA Similarity = Corresponding Sides of Similar Polygons are Proportional = Segment addition postulate + = + Definition of addition of fractions 1 + = 1 + Identity property of addition = Subtraction property of equality C DAEC BD BE E B A D BABC BD BE BD + DABE + EC BDBE BDDABEEC BD BD BE BE DAEC BD BE DAEC BD BE

3. Proportions & Similar Triangles Triangle Proportionality Converse • If a line divides the two sides of a triangle proportionally, then it is parallel to the third side • In the figure at right, if = then DE || AC • This is just the previous proof in reverse Statement Reason = Given = See previous proof ÐB ÐB Reflexive property of congruence ∆ABC ~ ∆DBE SAS Similarity ÐA ÐBDE Corresponding Angles of Similar Polygons are Congruent DE || AC Corresponding angles are congruent converse C E DAEC BD BE B A D DAEC BD BE BABC BD BE

4. Proportions & Similar Triangles • A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle • Is the smaller triangle similar to the original triangle? • What is the relationship between the length of the midsegment and the length of the third side? • The triangle is like a trapezoid with one base of length zero • Every triangle has three midsegments Triangle Midsegment Theorem • The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long Three Midsegments Theorem • The three midsegments of a triangle divide it into four congruent triangles

5. Right Triangle Similarity • The altitude to the hypotenuse has been constructed in each right triangle below • This construction creates two smaller right triangles within each original right triangle • Calculate the measures of the acute triangles in each diagram • How do the smaller right triangles compare in each diagram? • How do they compare to the original right triangle? Right Triangle Altitude Similarity Theorem • The altitude to the hypotenuse of a right triangle divides the triangle into two right triangles that are similar to each other and to the original triangle

6. Right Triangle Similarity • The altitude to the hypotenuse has been constructed in each right triangle below • By the previous theorem, the smaller triangles within each large triangle are similar to each other and to the large triangle • Complete each proportion for these right triangles: Right Triangle Altitude Ratio Theorem • The altitude (length h) to the hypotenuse of a right triangle divides the hypotenuse into two segments (of lengths p and q), such that • This means that the altitude h is the geometric meanof p and q = = = =