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Difficulty: MediumPolygon Angles and Properties

A designer is creating a custom floor tile in the shape of an irregular convex pentagon. The tile has one right angle. The remaining four interior angles are in the ratio 2:3:3:42:3:3:4. What is the measure, in degrees, of the largest interior angle of this tile?

  1. A
    112.5∘112.5^\circ
  2. B
    120∘120^\circ
  3. 150∘150^\circAnswer
  4. D
    180∘180^\circ
  5. E
    270∘270^\circ

Answer

The largest interior angle of the tile measures 150∘150^\circ.
The correct answer is 150∘150^\circ. The sum of the interior angles of a pentagon is 540∘540^\circ. Subtracting the right angle (90∘90^\circ) leaves 450∘450^\circ for the remaining four angles. Since they are in the ratio 2:3:3:42:3:3:4, their sum can be represented as 12y=450∘12y = 450^\circ, which yields y=37.5∘y = 37.5^\circ. The largest of these angles is 4y=150∘4y = 150^\circ, which is greater than the other angles (75∘75^\circ, 112.5∘112.5^\circ, and 90∘90^\circ).

Step-by-Step Solution

1
Calculate the sum of the interior angles of a pentagon.
The sum of the interior angles is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
A pentagon has 5 sides, and the sum of the interior angles of any convex nn-gon is given by the formula (n−2)×180∘(n - 2) \times 180^\circ.
2
Subtract the right angle to find the sum of the remaining four interior angles.
The sum of the remaining angles is 540∘−90∘=450∘540^\circ - 90^\circ = 450^\circ.
One of the angles is a right angle, which measures 90∘90^\circ.
3
Set up an equation using the given ratio to find the value of one part of the ratio.
Let the four remaining angles be 2y2y, 3y3y, 3y3y, and 4y4y. Their sum is 2y+3y+3y+4y=12y=450∘2y + 3y + 3y + 4y = 12y = 450^\circ, which gives y=37.5∘y = 37.5^\circ.
The remaining angles are in the ratio 2:3:3:42:3:3:4, so their measures are proportional to these values.
4
Calculate the measure of the largest interior angle.
The largest angle corresponds to the largest term in the ratio, which is 4y4y. Thus, the largest angle is 4×37.5∘=150∘4 \times 37.5^\circ = 150^\circ.
The question asks for the measure of the largest interior angle.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratios can be used to partition a total quantity into proportional parts.
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Topics

Polygon Angles and Properties

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