Question

1. the vertices of \\( \triangle lmn \\) are \\( l(7,4), m(7,16) \\), and \\( n(42,4) \\).

a. find the length of each side of the triangle.
b. what is the perimeter of the triangle?
c. what is the area of the triangle? explain how you determined your answer.

Explanation:

Step1: Find the length of \(LM\)

Since \(L(7,4)\) and \(M(7,16)\) have the same \(x -\)coordinate, use the distance formula for vertical distance \(d=\vert y_2 - y_1\vert\).
\(LM=\vert16 - 4\vert=12\)

Step2: Find the length of \(LN\)

Since \(L(7,4)\) and \(N(42,4)\) have the same \(y -\)coordinate, use the distance formula for horizontal distance \(d=\vert x_2 - x_1\vert\).
\(LN=\vert42 - 7\vert = 35\)

Step3: Find the length of \(MN\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(M(7,16)\) and \(N(42,4)\)
\(MN=\sqrt{(42 - 7)^2+(4 - 16)^2}=\sqrt{35^2+(- 12)^2}=\sqrt{1225 + 144}=\sqrt{1369}=37\)

Step4: Calculate the perimeter \(P\)

The perimeter of a triangle \(P=a + b + c\), where \(a = 12\), \(b = 35\), \(c = 37\)
\(P=12 + 35+37=84\)

Step5: Calculate the area \(A\)

Since \(LM\) and \(LN\) are perpendicular (one is vertical and one is horizontal), use the formula \(A=\frac{1}{2}\times base\times height\). Let \(base = LN = 35\) and \(height=LM = 12\)
\(A=\frac{1}{2}\times35\times12=210\)

Answer:

a. \(LM = 12\), \(LN = 35\), \(MN = 37\)
b. \(84\)
c. \(210\)