Recall that the sun's rays are parallel. Can we create any triangles using this fact?
30 feet
Practice makes perfect
We are told that a person who is 6 feet tall casts a 3-foot-long shadow and that a nearby pine tree casts a 15-foot-long shadow. We want to find the height of the pine tree h. First, let's draw a picture that illustrates the situation.
Now we draw the sun's rays in the picture. These parallel lines create the shadows cast by the person and the tree. We can also name the two triangles created by the sun's rays, the shadows, and the person or tree to make our calculations easier. Let's do it!
The shadows are perpendicular to the objects that cast them. This means that both triangles contain a right angle, so they are right triangles. Next, notice that ∠ ABC and ∠ DEF are corresponding angles. We know that if a transversal intersects parallel lines, then the corresponding angles are congruent. Since the sun's rays are parallel, ∠ ABC and ∠ DEF are congruent.
We found that two angles in △ ABC are congruent to two angles in △ DEF. Because of this, the third angles are also congruent, so the triangles are similar. Next, let's use the fact that corresponding sides in similar triangles have equivalent ratios to write a ratio to find the value of h.
AB/AC=DE/DF [0.3em]
⇕ [0.3em]
6/3=h/15
Let's solve for h.
