Question

triangle uvw, with vertices u(-3,4), v(-6,9), and w(-8,7), is drawn inside a rectangle, as shown below. what is the area, in square units, of triangle uvw?

Explanation:

Step1: Find the area of the rectangle

The length of the rectangle along the x - axis is from \(x=-8\) to \(x = - 3\), so the length \(l=|-3-(-8)| = 5\). The length along the y - axis is from \(y = 4\) to \(y = 9\), so the width \(w=|9 - 4|=5\). The area of the rectangle \(A_{r}=l\times w=5\times5 = 25\).

Step2: Find the areas of the three right - angled triangles inside the rectangle

For the first right - angled triangle with vertices \(U(-3,4)\), \(V(-6,9)\):
The base \(b_1=|-3-(-6)| = 3\) and the height \(h_1=|9 - 4|=5\). Its area \(A_1=\frac{1}{2}\times b_1\times h_1=\frac{1}{2}\times3\times5=\frac{15}{2}\).
For the second right - angled triangle with vertices \(V(-6,9)\), \(W(-8,7)\):
The base \(b_2=|-6-(-8)| = 2\) and the height \(h_2=|9 - 7|=2\). Its area \(A_2=\frac{1}{2}\times b_2\times h_2=\frac{1}{2}\times2\times2 = 2\).
For the third right - angled triangle with vertices \(W(-8,7)\), \(U(-3,4)\):
The base \(b_3=|-3-(-8)| = 5\) and the height \(h_3=|7 - 4|=3\). Its area \(A_3=\frac{1}{2}\times b_3\times h_3=\frac{1}{2}\times5\times3=\frac{15}{2}\).

Step3: Find the area of \(\triangle UVW\)

The area of \(\triangle UVW\) is \(A = A_{r}-A_1 - A_2 - A_3\).
\[

$$\begin{align*} A&=25-\frac{15}{2}-2-\frac{15}{2}\\ &=25-( \frac{15}{2}+\frac{15}{2})-2\\ &=25 - 15-2\\ &=8 \end{align*}$$

\]

Answer:

8