Question

What is the length of EF⎯⎯⎯⎯⎯?

Triangle ABC is similar to triangle DEF . The length of AC⎯⎯⎯⎯⎯ is 12 cm. The length of BC⎯⎯⎯⎯⎯ is 18 cm. The length of DF⎯⎯⎯⎯⎯ is 10 cm.

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Answer 1

`EF = "15 cm"`

Explanation:

If `triangleABC" ~ "triangle DEF`

then `(AB)/(DE)=(AC)/(DF)=(BC)/(EF)`.

In other words, all pairs of corresponding sides share the same scale factor. As an example, if `triangleABC` was twice as big as `triangleDEF`, then `bar(AB)` would be twice as long as `bar(DE).` That is: `(AB)/(DE)=2.` This ratio must then hold true for all pairs of corresponding sides, i.e. `(AC)/(DF)=2,` and `(BC)/(EF)=2.`

When we don't know the scale factor for two similar triangles, we can figure it out by computing the ratio of two corresponding sides. In this question, we know `AC` corresponds to `DF`, so the scale factor for these two triangles is

`(AC)/(DF)=("12 cm")/("10 cm")=6/5.`

Thus, `triangle ABC` is `6/5` as big as `triangle DEF.` And this scale factor must hold for all other pairs of corresponding sides, so:

`(BC)/(EF)=6/5`

And since we know `BC="18 cm",` we can plug that in:

`("18 cm")/(EF)=6/5`

Cross multiply:

`"18 cm" xx 5 = EF xx 6`

`"90 cm"/6 = EF`

`"15 cm" = EF`

Thus, `EF = "15 cm"`.