The length of the lighthouse shadow is: 61.2 ft
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To find the length of the lighthouse shadow given similar triangles, solve a proportion with corresponding sides.
Let's follow these steps:
In the diagram, LH is the lighthouse. TY is Tommy. HE is the lighthouse shadow. YE is Tommy's shadow. Notice that the green triangle, ΔTYE, fits onto the larger triangle, ΔLHE.
Using AA similarity, we know that the two triangles in the diagram are similar. To say the triangles are similar, write: ΔLHE ~ ΔTYE
Similar triangles have proportionate sides. This means that the length of corresponding sides will be longer or shorter by the same multiplication factor, which we can write as K.
Our pairs of corresponding sides are:
Corresponding sides are related to each other by the multiplication factor, K:
This makes any pair of corresponding sides written as a fraction equal to each other. Thus, we can write a proportion using these pairs.
We should choose two pairs of corresponding sides to write a proportion from:
[tex]\displaystyle { \frac{LH}{TY} = \frac{HE}{YE} }[/tex]
Note that this works because: [tex]\displaystyle { K = \frac{LH}{TY} = \frac{HE}{YE} }[/tex]
[tex]\displaystyle { \frac{LH}{TY} = \frac{HE}{YE} }[/tex] Write the proportion.
[tex]\displaystyle { \frac{170}{5} = \frac{HE}{1.8} }[/tex] Substitute known lengths.
[tex]\displaystyle { 34= \frac{HE}{1.8} }[/tex] Simplify. Now, isolate HE.
[tex]\displaystyle { 34*1.8 = \frac{HE}{1.8}*1.8}[/tex] Multiply both sides by 1.8.
[tex]\displaystyle { 34*1.8= HE}[/tex] 1.8 cancels out on the right side.
[tex]\displaystyle { HE = 34*1.8}[/tex] Keep the missing variable on the left side.
[tex]HE = 61.2[/tex] Solved for HE.
Remember to write the final answer with units.
Therefore, the lighthouse shadow is 61.2 ft long.
Learn more about AA similarity here: https://brainly.com/question/21247688
