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Difficulty: HardTriangle Properties and Angle Theorems

The measures of the interior angles of a triangle are in the ratio 2:3:72:3:7. If the measure of the largest angle is decreased by 15∘15^\circ and the measure of the smallest angle is increased by 15∘15^\circ, what is the ratio of the interior angles of the new triangle, ordered from smallest to largest?

  1. A
    1:2:31:2:3
  2. B
    1:3:81:3:8
  3. 1:1:21:1:2Answer
  4. D
    2:3:62:3:6
  5. E
    5:6:135:6:13

Answer

The ratio of the interior angles of the new triangle is 1:1:21:1:2.
The correct answer is the ratio 1:1:21:1:2. The sum of the interior angles in a triangle is always 180∘180^\circ. Given the ratio 2:3:72:3:7, the sum of the parts is 1212, which means each part represents 180∘/12=15∘180^\circ / 12 = 15^\circ. The original angles are therefore 30∘30^\circ, 45∘45^\circ, and 105∘105^\circ. Increasing the smallest angle by 15∘15^\circ gives 45∘45^\circ, and decreasing the largest by 15∘15^\circ gives 90∘90^\circ. The new angle measures are 45∘45^\circ, 45∘45^\circ, and 90∘90^\circ, which simplifies to 1:1:21:1:2.

Step-by-Step Solution

1
Set up the equation for the sum of the interior angles of a triangle.
2x+3x+7x=180∘2x + 3x + 7x = 180^\circ
The sum of the interior angles of any triangle is always 180∘180^\circ.
2
Solve for the value of xx.
12x=180∘  ⟹  x=15∘12x = 180^\circ \implies x = 15^\circ
Combining like terms simplifies the equation to find the value of one ratio unit.
3
Calculate the original measures of the three angles.
Smallest: 30∘30^\circ, Middle: 45∘45^\circ, Largest: 105∘105^\circ
Multiply each part of the ratio by x=15∘x = 15^\circ.
4
Apply the modifications to the smallest and largest angles.
New smallest: 30∘+15∘=45∘30^\circ + 15^\circ = 45^\circ; New largest: 105∘−15∘=90∘105^\circ - 15^\circ = 90^\circ; Middle: 45∘45^\circ (unchanged).
Perform the operations described in the problem statement.
5
Order the new angle measures from smallest to largest and simplify the ratio.
45∘:45∘:90∘  ⟹  1:1:245^\circ : 45^\circ : 90^\circ \implies 1 : 1 : 2
Divide each term in the ratio by the greatest common divisor, which is 4545.

Key Concept

Angle sum theorem of a triangle and ratio partition applications

Alternative Method

Instead of calculating the actual angle values, note that the sum of the ratio parts is 2+3+7=122+3+7 = 12, and the sum of the interior angles of a triangle is 180∘180^\circ. This means 11 ratio unit is equal to 180∘/12=15∘180^\circ / 12 = 15^\circ. Since the smallest angle is increased by 15∘15^\circ (exactly 11 ratio unit) and the largest is decreased by 15∘15^\circ (exactly 11 ratio unit), we can apply these modifications directly to the ratio terms. The new ratio terms are 2+1=32+1 = 3, 33 (unchanged), and 7−1=67-1 = 6. This gives a ratio of 3:3:63:3:6, which simplifies to 1:1:21:1:2.
Estimated Time:1m 30s
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Topics

Triangle Properties and Angle Theorems

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