What if your history teacher and your art teacher teamed up to give you an assignment that would count as a grade in both classes? Since you're learning about the U.S. government and life at the White House and also about drawing to scale, you are asked to create a scale drawing of the White House.
The following problems could appear as parts of the White House that you need to draw for your assignment. Study each slide carefully to see how proportions can be used to determine if figures are similar.
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Are the two quadrilaterals DCBA and PONM above similar? We can use proportions to check. The first step is to write the ratios for each of the corresponding sides to see if they are equal. \(\mathsf{ \frac{DC}{PO} = \frac{CB}{ON} = \frac{BA}{NM} = \frac{AD}{MP} }\) \(\mathsf{ \frac{6}{4} = \frac{3}{2} = \frac{9}{6} = \frac{6}{4} }\) All of these fractions are equal and reduce to a common ratio of \(\small\mathsf{ \frac{3}{2} }\). Since this is true, the two quadrilaterals are similar.
Are triangles ABC and DEF similar? Once again we can use proportions to check if these two triangles are congruent. \(\mathsf{ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} }\) \(\mathsf{ \frac{4}{24} = \frac{5}{30} = \frac{3}{18} }\) All these fractions are equal and reduce to a common fraction of \(\small\mathsf{ \frac{1}{6} }\). Therefore, these two triangles are similar.
Are the two triangles above similar? We can use proportions to check. First write out the corresponding sides as ratios: \(\mathsf{ \frac{18}{22} = \frac{16}{20} = \frac{20}{25} }\) Next reduce all the fractions to see if the proportions are equal. \(\mathsf{ \frac{9}{11} \neq \frac{4}{5} = \frac{4}{5} }\) Only two of the ratios are equal therefore these are not similar triangles. Now try this. If triangles IJK and LMN are similar, find the missing side x using proportions.
Since these two triangles are similar their sides have a common ratio. Therefore we can write the following proportion.
\(\mathsf{ \frac{JK}{MN} = \frac{IJ}{ML} }\) \(\mathsf{ \frac{54}{42} = \frac{x}{28} }\) Now you can cross multiply to find x. 54(28) = 42(x) 1512 = 42x Divide both sides by 42. x = 36 |
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