Question

what is the area of triangle qrs?
7 square units
9 square units
10 square units
13 square units

Explanation:

Step1: Find the base and height

Use the distance formula or count units. Assume base \( b = 6\) (from \(y = 2\) to \(y=-4\) vertically for a vertical - like side approximation, but better use the formula for area of a triangle with coordinates. Another approach: enclose the triangle in a rectangle. The rectangle has length \(4\) (from \(x=-2\) to \(x = 1\)) and height \(6\) (from \(y = 2\) to \(y=-4\)). Area of rectangle \(A_{rect}=4\times6 = 24\) square units.

Step2: Subtract the areas of the surrounding right - triangles

There are three right - triangles.
First triangle: base \(3\) (from \(x=-2\) to \(x = 1\)), height \(2\) (from \(y=-2\) to \(y = 2\)). Area \(A_1=\frac{1}{2}\times3\times2=3\) square units.
Second triangle: base \(3\) (from \(x=-2\) to \(x = 1\)), height \(2\) (from \(y=-2\) to \(y=-4\)). Area \(A_2=\frac{1}{2}\times3\times2 = 3\) square units.
Third triangle: base \(4\) (from \(y=-4\) to \(y = 2\)), height \(1\) (from \(x=-1\) to \(x=0\)). Area \(A_3=\frac{1}{2}\times4\times1=2\) square units.

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

Using the formula \(A = A_{rect}-A_1 - A_2 - A_3\).
\(A=24-(3 + 3+2)=24 - 8=9\) square units.

Answer:

9 square units