Question

what is the area of triangle lmn? 3 square units 4 square units 6 square units 8 square units

Explanation:

Step1: Calculate the area of rectangle

The rectangle formed by \(x = 1,x = 3,y=-1,y = 3\) has length \(l=3 - 1=2\) and width \(w = 3-(-1)=4\). Area of rectangle \(A_{rectangle}=l\times w=2\times4 = 8\) square units.

Step2: Calculate the area of non - triangle parts

There are two right - angled triangles.
For the triangle with vertices \(N(1,3),P(1,2),M(3,2)\): base \(b_1=3 - 1 = 2\), height \(h_1=3 - 2=1\). Area \(A_1=\frac{1}{2}\times b_1\times h_1=\frac{1}{2}\times2\times1 = 1\) square unit.
For the triangle with vertices \(L(1,-1),P(1,2),M(3,2)\): base \(b_2=3 - 1=2\), height \(h_2=2-(-1)=3\). Area \(A_2=\frac{1}{2}\times b_2\times h_2=\frac{1}{2}\times2\times3 = 3\) square units.

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

Using the formula \(A = A_{rectangle}-A_1 - A_2\). Substitute \(A_{rectangle}=8\), \(A_1 = 1\), \(A_2=3\) into the formula.
\(A=8-(1 + 3)=4\) square units.

Answer:

4 square units