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The lengths of the three sides of a triangle (in inches) are consecutive integers. If the perimeter is 69 inches, find the value of the shortest of the three side lengths. Answer: inches

The lengths of the three sides of a triangle (in inches) are consecutive integers. If the perimeter is 6969 inches, find the value of the shortest of the three side lengths.\newlineAnswer: inches

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Full solution

Q. The lengths of the three sides of a triangle (in inches) are consecutive integers. If the perimeter is 6969 inches, find the value of the shortest of the three side lengths.\newlineAnswer: inches
  1. Denote shortest side: Let's denote the shortest side of the triangle as xx inches. Since the sides are consecutive integers, the other two sides will be x+1x+1 inches and x+2x+2 inches. The perimeter of the triangle is the sum of the lengths of its sides.\newlinePerimeter = x+(x+1)+(x+2)x + (x + 1) + (x + 2)
  2. Perimeter equation setup: Given that the perimeter is 6969 inches, we can set up the equation:\newline69=x+(x+1)+(x+2)69 = x + (x + 1) + (x + 2)\newlineNow, we will combine like terms to simplify the equation.\newline69=3x+369 = 3x + 3
  3. Combine like terms: To find the value of xx, we need to isolate xx on one side of the equation. We will subtract 33 from both sides of the equation.69−3=3x+3−369 - 3 = 3x + 3 - 366=3x66 = 3x
  4. Isolate x: Now, we will divide both sides of the equation by 33 to solve for x.\newline663=3x3\frac{66}{3} = \frac{3x}{3}\newline22=x22 = x
  5. Check solution: We have found that the shortest side of the triangle is 2222 inches. To ensure there is no math error, we can check if the other two sides (2323 and 2424 inches) and the shortest side add up to the perimeter.\newline22+23+24=6922 + 23 + 24 = 69\newline69=6969 = 69\newlineThis confirms that our calculation is correct.

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