Question

which expression can be used to find the area of triangle rst?
(8·4)−(10 + 12 + 16)
(8·4)−\frac{1}{2}(10 + 12 + 16)
(8·4)−(5−6−8)
(8·4)−\frac{1}{2}(5 + 6 + 8)

Explanation:

Step1: Use the formula for area of triangle using coordinate system (shoelace formula alternative - area of rectangle minus areas of surrounding triangles)

The idea is to enclose the triangle \(RST\) in a rectangle. The area of the rectangle is \(l\times w\). Here, if we consider the rectangle that can enclose the triangle \(RST\), assume the length \(l = 8\) and width \(w=4\), so area of rectangle \(A_{rect}=8\times4\).
The area of a triangle can be found as \(A = A_{rect}-\sum A_{sur - \triangle}\). The area of a triangle is \(A=\frac{1}{2}bh\).
For the surrounding triangles:

• One triangle with base \(b_1 = 5\) and height \(h_1 = 4\) (area \(A_1=\frac{1}{2}\times5\times4 = 10\))
• Another triangle with base \(b_2 = 6\) and height \(h_2 = 4\) (area \(A_2=\frac{1}{2}\times6\times4=12\))
• The third triangle with base \(b_3 = 8\) and height \(h_3 = 4\) (area \(A_3=\frac{1}{2}\times8\times4 = 16\))

Step2: Calculate the area of \(\triangle RST\)

The area of \(\triangle RST\) is given by the formula \(A=(8\times4)-\frac{1}{2}(10 + 12+16)\)

Answer:

\((8\cdot4)-\frac{1}{2}(10 + 12 + 16)\)