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Zorluk: KolayTriangle Properties and Angle Theorems

A triangle has two sides of lengths 55 centimeters and 1111 centimeters. Which of the following could be the perimeter of the triangle, in centimeters?

  1. A
    16
  2. B
    22
  3. 27Cevap
  4. D
    32
  5. E
    34

Cevap

27
The correct answer is 2727 centimeters. According to the Triangle Inequality Theorem, the length of the third side, xx, of a triangle must be strictly greater than the difference of the other two sides (11−5=611 - 5 = 6) and strictly less than their sum (11+5=1611 + 5 = 16). This gives the inequality range 6<x<166 < x < 16. The perimeter is the sum of all three sides, which is P=5+11+x=16+xP = 5 + 11 + x = 16 + x. Applying the bounds of xx, we find that the perimeter must satisfy 16+6<P<16+1616 + 6 < P < 16 + 16, which simplifies to 22<P<3222 < P < 32. Among the options, 2727 is the only value that is strictly within this range.

Adım Adım Çözüm

1
Apply the Triangle Inequality Theorem to find the limits for the third side.
Let the third side be xx. The length of xx must satisfy: 11−5<x<11+5  ⟹  6<x<1611 - 5 < x < 11 + 5 \implies 6 < x < 16.
The length of any side of a triangle must be strictly between the positive difference and the sum of the lengths of the other two sides.
2
Set up the equation for the perimeter of the triangle.
Perimeter P=5+11+x=16+xP = 5 + 11 + x = 16 + x.
The perimeter of a triangle is defined as the sum of the lengths of all three of its sides.
3
Find the range of possible values for the perimeter PP by applying the inequality bounds of xx.
Add 1616 to all parts of the inequality 6<x<166 < x < 16: 16+6<16+x<16+16  ⟹  22<P<3216 + 6 < 16 + x < 16 + 16 \implies 22 < P < 32.
We must shift the inequality bounds of the third side by the sum of the two known sides to find the range of the perimeter.
4
Identify the option that falls strictly inside the range 22<P<3222 < P < 32.
The value 2727 is the only option that satisfies 22<27<3222 < 27 < 32.
Only a value strictly between 2222 and 3232 can represent a mathematically valid perimeter for this triangle.

Anahtar Kavram

Triangle Inequality Theorem and Perimeter
Tahmini Süre:1m 0s
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Konular

Triangle Properties and Angle Theorems

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