Exercises. Similar Triangles 1. Two triangles are similar if two sides and the median to one of these sides of one triangle are proportional to the corresponding parts of the other triangle. 2. Two right triangles are similar if the hypotenuse and a leg of the one are proportional to the hypotenuse and the corresponding leg of the other. HINT: Draw the median to the hypotenuse in each triangle. 302. Summary of Similar Triangles Two triangles are similar if a) two angles of the one are equal to two angles of the other; c) their sides are respectively proportional; d) their sides are respectively parallel; e) their sides are respectively perpendicular. The importance of condition (a) has been emphasized as Tool XXIII. 303. From the definition of similar polygons (§ 292) and the proofs of Propositions 1 and 3, the following Tool may be stated. Tool XXIV. (a) Corresponding angles of similar triangles are equal. (b) Corresponding sides of similar triangles are proportional. (c) Corresponding sides of similar triangles lie opposite equal angles. HOW AN AERIAL PHOTOGRAPH IS MADE (See photograph, page 214.) Before such a photograph is taken from an airplane, the line of flight, speed of the airplane, interval between exposures, and distance from the ground are calculated by the aid of geometry. Why are the successive pictures made to overlap? What triangles are congruent, similar, or equal? What lines are equal or parallel? Can you form a problem of your own based upon the lines in the picture? Given: P~ P'; g and g', h and h', etc., are corresponding diagonals making AR and R', AS and S', etc. To prove: AR~AR', AS AS', etc. ~ HINT: What facts are known about similar polygons? What is the usual Tool used to prove triangles similar? Proposition 5 (c N) 305. Theorem. Polygons formed by like-placed similar Given: P and P'; ARAR', AS~AS', etc. To prove: P ~ P'. HINT: What two conditions make polygons similar? What Facts are known about similar triangles? 3. 21 21', ≤2 = ≤2′, ≤3 = 43', etc. 4. = (2+3)= < (2′ + 3′), ≤ (4 +5)= < (4'5'), etc. 5. P ~ P'. Exercises. Similar Polygons FACTS 1. Given. 2. Corr. sides of similar triangles. 3. Why? 4. Why? 5. Definition. 1. Within quadrilateral ABCD join any point P to each of the vertices. From any point A' of PA draw a line parallel to AB intersecting PB in B'; from B' draw B'C' || BC; continue in like manner and finally join D' and A'. Prove ABCD ~ A'B'C'D'. 2. Prove the above conclusion if P is taken as any point without ABCD. |