What is the perimeter of trapezoid JKLM?\(\sqrt { 2 } + \sqrt { 5} \) units\(2+ \sqrt { 2 } + \sqrt { 5} \)units\(9+ 2\sqrt { 2} \)units\(9+ \sqrt { 2 } + \sqrt { 5} \)units

1. Calculate the lengths of the sides:

• JK:
  \[\sqrt { ( ( - 4) - ( - 7) ) ^ 2 + ( 4 - 4) ^ 2} = \sqrt { ( 3) ^ 2 + 0} = 3 \text { units} \]
• LM:
  \[\sqrt { ( ( - 2) - ( - 8) ) ^ 2 + ( 3 - 3) ^ 2} = \sqrt { ( 6) ^ 2 + 0} = 6 \text { units} \]
• JM:
  \[\sqrt { ( ( - 8) - ( - 7) ) ^ 2 + ( 3 - 4) ^ 2} = \sqrt { ( - 1) ^ 2 + ( - 1) ^ 2} = \sqrt { 1 + 1} = \sqrt { 2} \text { units} \]
• KL:
  \[\sqrt { ( ( - 2) - ( - 4) ) ^ 2 + ( 3 - 4) ^ 2} = \sqrt { ( 2) ^ 2 + ( - 1) ^ 2} = \sqrt { 4 + 1} = \sqrt { 5} \text { units} \]

Sum the lengths to find the perimeter:
\[JK + LM + JM + KL = 3 + 6 + \sqrt { 2} + \sqrt { 5} = 9 + \sqrt { 2} + \sqrt { 5} \text { units} \]
Thus, the perimeter of trapezoid JKLM is \(9 + \sqrt { 2} + \sqrt { 5} \) units.

Supplemental Knowledge

The perimeter of a trapezoid is the sum of the lengths of all its sides. To find the perimeter, you need to calculate the distance between each pair of vertices.
For a trapezoid with vertices \(J( x_ 1, y_ 1) \), \(K( x_ 2, y_ 2) \), \(L( x_ 3, y_ 3) \), and \(M( x_ 4, y_ 4) \), you can use the distance formula to find the length of each side:
\[d = \sqrt { ( x_ 2 - x_ 1) ^ 2 + ( y_ 2 - y_ 1) ^ 2} \]

Real-Life Connections

Understanding how to calculate perimeters is useful in various real-world applications:

• Construction and Landscaping: Knowing the perimeter helps in determining the amount of materials needed for fencing or bordering.
• Interior Design: Calculating perimeters assists in planning layouts and ensuring that furniture fits within designated spaces.

Imagine designing your garden around a trapezoidal flower bed. By accurately calculating its perimeter, you're able to purchase just enough edging material to surround it perfectly and complete its outline.

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