Question

1. triangle abc

a(2,2)
b(5,7)
c(8,2)
perimeter
area

Explanation:

Step1: Calculate the length of \(AB\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(A(2,2)\) and \(B(5,7)\), \(x_1 = 2,y_1 = 2,x_2 = 5,y_2 = 7\).
\(AB=\sqrt{(5 - 2)^2+(7 - 2)^2}=\sqrt{9 + 25}=\sqrt{34}\approx5.83\)

Step2: Calculate the length of \(BC\)

For \(B(5,7)\) and \(C(8,2)\), \(x_1 = 5,y_1 = 7,x_2 = 8,y_2 = 2\).
\(BC=\sqrt{(8 - 5)^2+(2 - 7)^2}=\sqrt{9+25}=\sqrt{34}\approx5.83\)

Step3: Calculate the length of \(AC\)

For \(A(2,2)\) and \(C(8,2)\), \(x_1 = 2,y_1 = 2,x_2 = 8,y_2 = 2\).
\(AC=\sqrt{(8 - 2)^2+(2 - 2)^2}=\sqrt{36+0}=6\)

Step4: Calculate the perimeter

Perimeter \(P=AB + BC+AC\approx5.83+5.83 + 6=17.66\)

Step5: Calculate the area

Since \(AC\) is parallel to the \(x\) - axis (\(y\) - coordinates of \(A\) and \(C\) are the same), and the height \(h\) from \(B\) to \(AC\) is the difference in \(y\) - coordinates (\(h = 7 - 2=5\)), base \(b = AC = 6\).
Using the formula \(A=\frac{1}{2}\times b\times h\), \(A=\frac{1}{2}\times6\times5 = 15\)

Answer:

Perimeter: \(17.66\), Area: \(15\)