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Question

Difficulty: EasyTriangle Properties and Angle Theorems

A triangle has two sides of length 77 centimeters and 1010 centimeters. If the length of the third side, in centimeters, must be an integer, what is the minimum possible length of the third side?

  1. A
    3
  2. B
    17
  3. 4Answer
  4. D
    16
  5. E
    8

Answer

4 centimeters
The Triangle Inequality Theorem states that the length of the third side, xx, must be strictly greater than the difference of the two known sides (10−7=310 - 7 = 3) and strictly less than their sum (10+7=1710 + 7 = 17). Therefore, 3<x<173 < x < 17. The smallest integer value that satisfies this inequality is 4.

Step-by-Step Solution

1
Identify the given side lengths of the triangle.
The two given side lengths are 7 centimeters and 10 centimeters.
These are the values needed to apply the Triangle Inequality Theorem.
2
Apply the Triangle Inequality Theorem to find the range of possible lengths for the third side, xx.
10−7<x<10+710 - 7 < x < 10 + 7, which simplifies to 3<x<173 < x < 17.
The theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.
3
Identify the minimum integer value within the valid range.
The smallest integer strictly greater than 3 is 4.
The question specifies that the third side length must be an integer and asks for the minimum possible value.

Key Concept

The Triangle Inequality Theorem states that for any triangle with sides aa, bb, and cc, the length of the third side must satisfy ∣a−b∣<c<a+b|a - b| < c < a + b.
Estimated Time:1m 0s
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Topics

Triangle Properties and Angle Theorems

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