Question 11·Medium·Lines, Angles, and Triangles
A flagpole that is 18 feet tall casts a 6-foot shadow on level ground at a certain time of day. At the same time, a nearby signpost casts a 1.5-foot shadow. How tall, in feet, is the signpost?
For shadow and height problems, immediately recognize that similar right triangles are formed and set up a proportion: . Simplify the known ratio first to get a clean multiplier (like ), then apply that multiplier to the unknown object’s shadow length to get its height, minimizing algebra and reducing calculation errors.
Hints
Notice the similar right triangles
Each object (flagpole and signpost) with its shadow and the ground forms a right triangle, and the sun’s rays are at the same angle for both. Think about what that means for the ratios of corresponding sides.
Relate height to shadow length
Focus on the ratio for the flagpole and set it equal to the same ratio for the signpost. Write this as a proportion.
Solve the proportion
Once you have , simplify the left side and then solve for by isolating it. What do you multiply by to match the simplified ratio?
Desmos Guide
Compute the height using the ratio
In Desmos, type 18/6*1.5 or (18/6)*(1.5) and look at the numeric result. This value is the height of the signpost in feet, found by applying the same height-to-shadow ratio to the -foot shadow.
Step-by-step Explanation
Set up the proportion using similar triangles
The sun hits both the flagpole and the signpost at the same angle, so the triangles formed by each object and its shadow are similar.
That means the ratio of height to shadow length is the same for both:
Substitute the known values:
where is the height of the signpost in feet.
Simplify the ratio from the flagpole
Simplify the left side of the proportion:
So the equation becomes:
This tells you that the signpost’s height is 3 times its shadow length.
Solve for the signpost’s height
Use the equation to solve for by multiplying both sides by :
So, the signpost is feet tall.