SIMILAR FIGURES. 205. When are two figures equivalent? equal? May a square equal a triangle? a rectangle? (1) Equal figures are not only equal in area, but they are similar in form; as we have learned, they can be made to coin. cide in all their parts. (2) Equivalent figures are equal in area, but they are not necessarily similar in form. (3) Similar polygons are mutually equiangular, and their homologous sides are proportional? Draw any and assume any side as the base; then draw a line which is parallel to the base and which cuts off of one side. How does it cut the other side? Note the two triangles. Are they equiangular? [?] Are their sides proportional? [?] Are they similar? 4. Homologous parts, sides, angles, etc., are those which are similarly situated. 5. Similar arcs subtend equal central angles; similar sectors and segments are those whose arcs subtend equal central angles. Are all circles similar? 206. PROPOSITION XIX. (1) Draw any triangle, assume any side the base, and fix any point in either of the sides. Let the be A B C, and A B the base, and P the point fixed in A C. Now draw a line through P which cuts BC at D and makes the CPD / CAB. Is PD parallel to the base? Can you prove the triangles similar? = (2) Given the triangles A B C and D E F in which angles A, B, and C are respectively equal to angles D, E, and F. Can you prove the triangles similar? Can you prove two triangles similar which are mutually equiangular? [See (2) above.] What two properties have similar polygons? [See $205, 3.] [Hint.-Superpose.] If you fail, after using the above exercises and hints as helps, see the further hint given below. [Hint.-Superpose, placing an upon its equal and similar side upon similar side. Show that sides are parallel; ... sides are proportional.] 207. Cor. I. Given any two triangles in which two angles of one equal two angles of the other. Prove the triangles similar. 208. Cor. II. Given two rt. triangles in which an acute of one equals an acute angle of the other. Prove the triangles similar. 209. Cor. III. Given two triangles similar to a third triangle. Make deduction and prove it. EXERCISES. 144. Given two unequal lines, a and b. Construct two squares having these lines respectively for sides. Can you prove these squares similar? Make general conclusion. 145. Given two equilateral triangles having unequal sides Can you prove them similar? 146. Given two isosceless having vertical angles equal. Can you prove them similar? 147. Given two regular hexagons. Prove that they are similar. 148. Given two regular polygons, each having n sides. Prove that they are similar. 210. PROPOSITION XX. You have proved in Prop. XIX. that if two triangles are mutually equiangular, the corresponding sides are proportional, and consequently the triangles are similar. Now can you prove that two triangles are similar if the sides of one are proportional to the sides of the other, each to each? This proof is difficult, so do not get discouraged if you fail. But make a stubborn fight before you give it up. Review all the suggestions given to pupils for attacking original demonstrations. If, having exhausted all your powers, you fail, consult the "hint," prove the Prop. yourself without further reading just as soon as you discover the proof, and compare your proof with the rest of the "hint." Possibly, part of your proof will be original. Try to see just what effort you failed to put forth in your trying to discover the demonstration. This may give success to future efforts. : [Hint.-Given the two triangles A B C and D E F, in which A B D E = BC: E FAC: DF. (Pupil may draw the As having sides 4", 5", and 6′′, and 6′′, 73′′, and 9′′, respectively. Make the sides of A B C longer than those of DEF.) On C A measure off C G FD and on C B meas ure off C H FE. Draw G H. (Now try to prove As CGH and C A B similar. Then try to prove s C G H and D E F equal.) (1) AC: DF = B C : E F. [?] (2) Then A C: GCBC: HC. [?] [?] (Compare (4) and (6) and make deductions.) (7) .. G H D E. [?] (8) And As G C H and D E F are equal. [?] (9) Also As A B C and D E F are similar, Q. E. D.] We have seen that if triangles are equiangular, their sides must be proportional, and conversely. Would you form the same conclusion about equiangular quadrilaterals and other equiangular polygons? (1) Compare a square with a rectangle, a rhombus with a rhomboid. (Draw figures.) (2) Also compare a square with a rhombus, a rectangle with a rhomboid. (Their sides may be equal or proportional.) (3) What conclusion do you reach? 211. PROPOSITION XXI. Draw two As A B C and D E F, making A B = 10′′, AC 8", B C 6", and DE 5", D F = 4", E F 3'. = = Are the As similar? Why? = - Draw the altitudes upon A B and D E. Compare them. Do they have the same ratio as A B and D E? as A C and D F? as BC and E F? Draw the other altitudes and compare their ratio with that of any two homologous sides. Draw other similar triangles and make further comparisons of the ratio of similar altitudes to the ratio of homologous sides. What deduction can you make of the altitudes of all similar triangles? Write Prop. XXI. and its formal proof. What is the converse of Prop. XXI? Can you prove it? EXERCISES. 149. In a city are two rectangular lots, one is 50 feet by 150 feet, and the other is 100 feet by 250 feet. Are they similar? Why? 150. Draw any scalene triangle. Can you draw an isosceles triangle similar to the triangle drawn? Prove your answer. 212. PROPOSITION XXII. Problem. To divide a line into parts proportional to two or more given lines. There are two equal fractions whose numerators are 3 and 4, respectively, and the sum of whose denominators is 49. Required the denominators. (1) Solve by Algebra. (2) Solve by Geometry. [Hint.-Use lines to represent the numbers.] [Hint.—If you fail on (2), draw any scalene ▲ and fix a point in one side. Note how the point divides the side. Now, how can you divide the other side into parts proporional to these two?] There are two equal fractions, the first is and the numerator of the second is 3. Required the denominator of the second. (1) Solve by Algebra. |