Questions: In the diagram below, EF is parallel to BC. If EF is 5 less than DF, DC=32, and BC=24, find the length of DF. Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.

In the diagram below, EF is parallel to BC. If EF is 5 less than DF, DC=32, and BC=24, find the length of DF. Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.

Transcript text: In the diagram below, $\overline{E F}$ is parallel to $\overline{B C}$. If $E F$ is 5 less than $D F, D C=32$, and $B C=24$, find the length of $\overline{D F}$. Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.

Solution

Solution Steps

Step 1: Setting up the proportion

Since EF is parallel to BC, triangles DEF and DBC are similar. Therefore, the ratio of corresponding sides is equal:

DF/DC = EF/BC

Step 2: Substituting the given values

We are given that DC = 32, BC = 24, and EF = DF - 5. Substituting these values into the proportion, we get:

DF/32 = (DF - 5)/24

Step 3: Cross-multiplying and solving for DF

Cross-multiply to get rid of the fractions:

24 * DF = 32 * (DF - 5)

24DF = 32DF - 160

160 = 8DF

DF = 20

Final Answer: The length of DF is 20.

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