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  1. The perimeter of a triangle is $9 x+2$. Two sides of the triangle are $4 x-5$ and $2 x+3$. Find the length of the third side.


  1. The side lengths of a rectangle are represented by the expressions $x+2$ and $x-5$. If the area of the rectangle is 60 square units, what is the perimeter?

Verified Answer :

  • Subtract the sum of the two given sides from the total perimeter to find the third side length of the triangle: \(s_3 = 3x + 4\).
  • Solve the quadratic equation \((x + 2)(x - 5) = 60\) to find the variable \(x\).
  • Choose the positive solution \(x = 10\) to ensure side lengths are valid, then calculate the perimeter of the rectangle as \(34\).
  • The final results are \(3x + 4\) for the third side and \(34\) for the perimeter. Final answer: \(\boxed{3x+4, 34}\).

Explanation

  1. Calculating the Triangle's Third Side

The triangle perimeter is given as \(P_T = 9x + 2\), and two side lengths are \(s_1 = 4x - 5\) and \(s_2 = 2x + 3\). The sum of all three sides equals the perimeter, so \(s_1 + s_2 + s_3 = P_T\). To find the third side \(s_3\), we rearrange this to \(s_3 = P_T - (s_1 + s_2)\). Substituting the given expressions, we have:
\[s_3 = (9x + 2) - ((4x - 5) + (2x + 3))\]
\[s_3 = (9x + 2) - (6x - 2)\]
\[s_3 = 9x + 2 - 6x + 2\]
\[s_3 = 3x + 4\]
For the triangle to exist, all sides must be positive, which implies \(4x - 5 > 0\), or \(x > 1.25\). Thus, the third side length is \(3x + 4\).

  1. Finding the Value of x for the Rectangle

The rectangle has side lengths \(l = x + 2\) and \(w = x - 5\), with an area \(A = 60\). The area is calculated as the product of the length and width, so \((x + 2)(x - 5) = 60\). Expanding the left side gives:
\[x^2 - 5x + 2x - 10 = 60\]
\[x^2 - 3x - 10 = 60\]
\[x^2 - 3x - 70 = 0\]
We solve this quadratic equation by factoring: \((x - 10)(x + 7) = 0\). This gives two potential values for \(x\): \(x = 10\) or \(x = -7\). Since the side length \(w = x - 5\) must be positive, \(x\) must be greater than \(5\). Therefore, we choose \(x = 10\).

  1. Calculating the Rectangle's Perimeter

Now that we have \(x = 10\), we calculate the rectangle's perimeter \(P_R = 2(l + w)\). Substituting the expressions for \(l\) and \(w\):
\[P_R = 2((x + 2) + (x - 5))\]
\[P_R = 2(2x - 3)\]
Substituting \(x = 10\):
\[P_R = 2(2(10) - 3) = 2(20 - 3) = 2(17) = 34\]
The perimeter of the rectangle is 34 units.

Examples

Geometric properties are essential for construction and design. For example, if a contractor knows the perimeter of a triangular plot and the lengths of two fence sections, they can determine the required length for the third fence. Similarly, calculating the perimeter of a rectangular room from its area allows for determining the amount of baseboard trim needed to finish the floor. Understanding these relationships ensures resources are estimated accurately.

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