Question

amaury draws an acute triangle on the coordinate grid. he finds the area of the triangle by drawing a rectangle that circumscribes the triangle and forming right triangles. show the steps amaury must take to find the area of the triangle. what is the area of the triangle in square units?

Explanation:

Step1: Determine the rectangle's dimensions

The rectangle that circumscribes the triangle has length from \(x = - 3\) to \(x=3\) (length \(l=6\)) and from \(y = 3\) to \(y = 7\) (width \(w = 4\)). Area of rectangle \(A_{r}=l\times w=6\times4 = 24\)

Step2: Calculate the area of the three right - angled triangles

• Triangle 1: Vertices \((-3,5)\), \((3,5)\), \((3,7)\). Base \(b_1=6\), height \(h_1 = 2\). Area \(A_1=\frac{1}{2}\times6\times2=6\)
• Triangle 2: Vertices \((-3,5)\), \((2,5)\), \((2,3)\). Base \(b_2 = 5\), height \(h_2=2\). Area \(A_2=\frac{1}{2}\times5\times2 = 5\)
• Triangle 3: Vertices \((2,3)\), \((3,3)\), \((3,7)\). Base \(b_3=1\), height \(h_3 = 4\). Area \(A_3=\frac{1}{2}\times1\times4=2\)

Step3: Find the area of the acute triangle

Area of acute triangle \(A=A_{r}-(A_1 + A_2+A_3)\)
\(A = 24-(6 + 5+2)\)
\(A=24 - 13\)

Answer:

\(11\) square units