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Find the ratio of the length of a diagonal and a side of a regular pentagon.
Two polygons are similar if corresponding angles are congruent and corresponding sides are proportional. For triangles, the congruence of two angles already implies similarity.
The Grim Reaper, who is 5 feet tall, stands 16 feet away from a street lamp at night. The Grim Reaper's shadow cast by the streetlamp light is 8 feet long. How tall is the street lamp?
Both the lamp post and the Grim Reaper stand vertically on horizontal ground.
A sketch of the situation is helpful for finding the solution. Under the assumption that the lamp post and the Grim Reaper make right angles in relation to the ground, two right triangles can be drawn. The unknown height of the lamp post is labeled as x.
As these triangles both have a right angle and share the angle on the right-hand side, they are similar by the Angle-Angle (AA) Similarity Theorem. Notice that the base of the larger triangle measures to be 24 feet.
Since the triangles are similar, the ratios between corresponding side lengths are the same.The street lamp at 15 feet high towers over The Grimp Reaper.
For the given diagram, find the missing length.
A second theorem allows for determining triangle similarity when only the lengths of corresponding sides are known.
Two theorems have been covered, now a third theorem that can be used to prove triangle similarity will be investigated. This third theorem allows for determining triangle similarity when the lengths of two corresponding sides and the measure of the included angles are known.
Through applying the theorems of similar triangles, the ratio of the lengths of a diagonal and the sides of a regular pentagon can be found.
Begin by determining the angle measures of the figure.
Substitute values
LHS−(36+36)=RHS−(36+36)
Next, focus on △ACE. In this triangle, AC and EC are diagonals of the pentagon, and AE is a side.
In the diagram, a smaller triangle labeled △AFE is also present. These two triangles share a common angle at A and congruent angles at C and E.LHS⋅(x−1)=RHS⋅(x−1)
Multiply parentheses
LHS−1=RHS−1
Use the Quadratic Formula: a=1,b=-1,c=-1
-(-a)=a
Calculate power and product
a−(-b)=a+b
Add terms
Length of sideLength of diagonal=21+5≈1.618
The ratio of the diagonal to the side of a regular pentagon can be used to prove that the following construction creates a regular pentagon. This is a construction created by Yosifusa Hirano in the 19th century.