Question

1. find the area of \\( \triangle a b c \\) with vertices \\( a ( - 7, - 5 ), b ( 1, - 5 ) \\), and \\( c ( - 7,1 ) \\).

a. 10 square units
b. 18 square units
c. 48 square units
d. 24 square units

1. find the perimeter of the figure:

a. 123 feet
b. 246 feet
c. 7252 feet
d 3626 feet

Explanation:

Step1: Calculate the length of \(AB\)

For points \(A(-7,-5)\) and \(B(1,-5)\), since the \(y -\)coordinates are the same, use the distance formula \(d=\vert x_2 - x_1\vert\).
\(AB=\vert1-(-7)\vert=\vert1 + 7\vert = 8\)

Step2: Calculate the length of \(AC\)

For points \(A(-7,-5)\) and \(C(-7,1)\), since the \(x -\)coordinates are the same, use the distance formula \(d=\vert y_2 - y_1\vert\).
\(AC=\vert1-(-5)\vert=\vert1 + 5\vert=6\)

Step3: Calculate the area of \(\triangle ABC\)

Since \(\triangle ABC\) is a right - triangle ( \(AB\perp AC\) as one is horizontal and the other is vertical), use the formula \(A=\frac{1}{2}\times base\times height\).
\(A=\frac{1}{2}\times AB\times AC=\frac{1}{2}\times8\times6 = 24\)

Step1: Recall the perimeter formula for a rectangle

The perimeter formula for a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width.

Step2: Substitute the values of \(l = 74\) ft and \(w = 49\) ft into the formula

\(P=2(74 + 49)\)
First, calculate the sum inside the parentheses: \(74+49 = 123\)
Then, multiply by 2: \(P=2\times123=246\)

Answer:

D. 24 square units