Answer:
The area of the triangle is 5.875 square units.
Step-by-step explanation:
First, we need to calculate the lengths of the triangle by means of Pythagorean Theorem:
[tex]a = \|\overrightarrow {QR}\| = \sqrt{(5-4)^{2}+(7-4)^{2}+(-2+1)^{2}}[/tex]
[tex]a \approx 3.317[/tex]
[tex]b = \|\overrightarrow {RS}\| = \sqrt{(3-5)^{2}+(2-7)^{2}+(-4+2)^{2}}[/tex]
[tex]b \approx 5.745[/tex]
[tex]c = \|\overrightarrow {QS}\| = \sqrt{(3-4)^{2}+(2-4)^{2}+(-4+1)^{2}}[/tex]
[tex]c \approx 3.742[/tex]
Now, we can use the following formula to fid the area of the triangle ([tex]A[/tex]), measured in square units:
[tex]A = \sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}[/tex]
Where:
[tex]s[/tex] - Semiperimeter of the triangle, dimensionless.
[tex]a[/tex], [tex]b[/tex], [tex]c[/tex] - Sides of the triangle, dimensionless.
The semiperimeter can be determined by this:
[tex]s = \frac{a+b+c}{2}[/tex]
If we know that [tex]a \approx 3.317[/tex], [tex]b \approx 5.745[/tex] and [tex]c \approx 3.742[/tex], the area of the triangle is:
[tex]s = \frac{3.317+5.745+3.742}{2}[/tex]
[tex]s = 6.402[/tex]
[tex]A = \sqrt{(6.402)\cdot (6.402-3.317)\cdot (6.402-5.745)\cdot (6.402-3.742)}[/tex]
[tex]A \approx 5.875[/tex]
The area of the triangle is 5.875 square units.
