Geometry Notes SIMILAR TRIANGLES


 Clyde Williamson
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1 Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving similr tringles. Vocbulry: As you re, you shoul be looking for the following vocbulry wors n their efinitions: similr tringles Formuls: You shoul be looking for the following formuls s you re: proportions for similr tringles We will continue our stuy of geometry by looking t similr tringles. Two tringles re similr is their corresponing ngles re congruent (or hve the sme mesurement). similr c f b e In the picture bove, the corresponing ngles re inicte in the two tringles by the sme number of hsh mrks. In other wors, the ngle with one hsh mrk in the smller tringle correspons to the ngle with one hsh mrk in the lrger tringle. These ngles hve the sme mesure or re congruent. There re lso corresponing sies in similr tringles. In the tringles bove sie correspons to sie (for exmple). These sies o not necessrily hve the sme mesure. They o, however, form rtio tht is the sme no mtter which pir of corresponing sies the rtio is me from. Thus we cn write the following eqution
2 Similr Tringles Pge 2 of 6 b c e f Notice tht the sies of one prticulr tringle re lwys written on top of the frctions n the sies of the other tringle re lwys written on the bottom of the frctions. It oes not mtter which tringle is put in which prt of the frction s longs s we re consistent within problem. Similr Tringle Rtios b c e f c f b e It shoul be note tht lthough our tringles re in the sme reltive position, this is not neee for tringles to be similr. One of the tringles cn be rotte or reflecte. Plese note tht pictures below re not rwn to scle. Exmple 1: Given tht the tringles re similr, fin the lengths of the missing sies. x y Solution: There is one sie missing in the tringle on the left. This sie is lbele x. There is lso one sie missing from the tringle on the right. This sie is lbele y. The sie x in the tringle on the left correspons to the sie lbele 10 in the tringle on the right. We
3 Similr Tringles Pge 3 of 6 know this becuse these sies connect the ngle with one hsh mrk to the ngle with three hsh mrks in ech of the tringles. The sie 100 in the tringle on the left correspons to the sie lbele 8 in the tringle on the right. Finlly the sie 90 in the tringle on the left correspons to the sie lbele y in the tringle on the right. We will nee this informtion to fin vlues for x n y. Fin x: We will strt by fining the vlue for x. We will nee to form rtio with the sie lbele x n is corresponing sie in the tringle on the right. We will lso nee pir of corresponing sie both of whose mesurements we hve n mke rtio out of those. Once we mke n eqution using these rtios, we will just nee to solve the eqution for x. I m going to choose to plce the sies of the tringle on the left on the top of ech of the rtios. x ( x ) 10(100) 8x 1000 x 125 units Fin y: We will continue by fining the vlue for y. We will nee to form rtio with the sie lbele y n is corresponing sie in the tringle on the right. We will lso nee pir of corresponing sie both of whose mesurements we hve n mke rtio out of those. Although we know the mesurements of ll the other sies of the tringle, it is best to voi using sie we just clculte to mke further clcultions if possible. The reson for this is tht if we hve me mistke in our previous clcultion, we woul en up with n incorrect nswer here s well. Once we mke n eqution using these rtios, we will just nee to solve the eqution for y. I m going to choose to plce the sies of the tringle on the left on the top of ech of the rtios.
4 Similr Tringles Pge 4 of y 8 8(90) y (100) y 7.2 units y Exmple 2: Given tht the tringles re similr, fin the lengths of the missing sies. y 45. x Solution: Our first job here is to etermine which sies in the tringle on the left correspon to which sies in the tringle on the right. The sie in the tringle on the left lbele y joins the ngle with one hsh mrk n the ngle with two hs mrks. The sie in the tringle on the right tht oes this is the sie lbele Thus these re corresponing sies. In similr mnner, we cn see tht the sie lbele 3.2 in the tringle on the left correspons to the sie lbele x on the tringle on the right. Finlly the sie lbele 2.8 in the tringle on the left correspons to the sie lbele 45 in the tringle on the right. Fin x: We will strt by fining the vlue for x. We will nee to form rtio with the sie lbele x n is corresponing sie in the tringle on the right. We will lso nee pir of corresponing sie both of whose mesurements we hve n mke rtio out of those. Once we mke n eqution using these rtios, we will just
5 Similr Tringles Pge 5 of 6 nee to solve the eqution for x. I m going to choose to plce the sies of the tringle on the left on the top of ech of the rtios x (45) x (2.8) x 144 units 2.8 This is pproximtely units. Fin y: We will continue by fining the vlue for y. We will nee to form rtio with the sie lbele y n is corresponing sie in the tringle on the right. We will lso nee pir of corresponing sie both of whose mesurements we hve n mke rtio out of those. Once we mke n eqution using these rtios, we will just nee to solve the eqution for y. I m going to choose to plce the sies of the tringle on the left on the top of ech of the rtios. y y (45) 37.25(2.8) 45y y units 45 This is pproximtely units. x Our finl exmple is n ppliction of similr tringles Exmple 3: A 3.6foot tll chil csts show of 4.7 feet t the sme instnt tht telephone pole csts show of 15 feet. How tll is the telephone pole? Solution: For this problem, it helps to hve picture.
6 Similr Tringles Pge 6 of 6 chil 3.6 feet pole show 4.7 feet show 15 feet When we look t the picture bove, we see tht we hve two tringles tht re similr. We re looking for the height of the pole. The pole sie of the tringle on the right correspons to the sie of the chil sie of the tringle on the left. The two show sies of the tringles lso correspon to ech other. Thus we cn write n eqution of rtios to fin the height of the pole pole 15 (3.6)(15) pole(4.7) pole 54 pole 4.7 Thus the pole is pproximtely feet high.
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