Question

△car has the following vertices: c(3, -2), a(1, 8), and r(-7, 3). use the herons formula to find the area of △car. round your answer to the nearest whole number. use the keypad to enter the answer in the box provided. the area of △car is square units.

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• For \(CA\): \(x_1 = 3,y_1=-2,x_2 = 1,y_2 = 8\)

\(CA=\sqrt{(1 - 3)^2+(8+ 2)^2}=\sqrt{(-2)^2+10^2}=\sqrt{4 + 100}=\sqrt{104}\)

• For \(AR\): \(x_1 = 1,y_1 = 8,x_2=-7,y_2 = 3\)

\(AR=\sqrt{(-7 - 1)^2+(3 - 8)^2}=\sqrt{(-8)^2+(-5)^2}=\sqrt{64 + 25}=\sqrt{89}\)

• For \(CR\): \(x_1 = 3,y_1=-2,x_2=-7,y_2 = 3\)

\(CR=\sqrt{(-7 - 3)^2+(3 + 2)^2}=\sqrt{(-10)^2+5^2}=\sqrt{100+25}=\sqrt{125}\)

Step2: Calculate the semi - perimeter \(s\)

\(s=\frac{\sqrt{104}+\sqrt{89}+\sqrt{125}}{2}\approx\frac{10.2+9.4+11.2}{2}=\frac{30.8}{2}=15.4\)

Step3: Apply Heron's formula \(A=\sqrt{s(s - a)(s - b)(s - c)}\)

\(A=\sqrt{15.4(15.4-\sqrt{104})(15.4-\sqrt{89})(15.4-\sqrt{125})}\)
\(A=\sqrt{15.4(15.4 - 10.2)(15.4 - 9.4)(15.4 - 11.2)}\)
\(A=\sqrt{15.4\times5.2\times6\times4.2}\)
\(A=\sqrt{15.4\times5.2\times25.2}\)
\(A=\sqrt{15.4\times131.04}\)
\(A=\sqrt{2018.016}\approx45\)

Answer:

\(45\)