Answer:
Step-by-step explanation:
The area for a trapezoid is
[tex]A=\frac{1}{2}h(b_{1}+b_{2})[/tex]
h is the length of ST, one of the bases is the length of MK, and the other base is the length of AS. First we'll find h:
The coordinates for S are (0, -2) and T are (1, 2). Using the distance formula:
[tex]d=\sqrt{(1-0)^2+(2-(-2))^2[/tex] and
[tex]d=\sqrt{17}[/tex]. So h = √17
Now for the length of MK. The coordinates for M are (-7, 4) and for K (5, 1). Using the distance formula again:
[tex]d=\sqrt{(-7-5)^2+(4-1)^2}[/tex] and
[tex]d=\sqrt{(-12)^2+(3)^2}[/tex] so
[tex]d=\sqrt{153}[/tex] which simplifies to
[tex]d=3\sqrt{17}[/tex]. So MK = 3√17.
Now for the length of AS. The coordinates for A: (-4, -1) and for S: (0, -2). Using the distance formula one more time:
[tex]d=\sqrt{(-4-0)^2+(-1-(-2))^2}[/tex] and
[tex]d=\sqrt{(-4)^2+(1)^2}[/tex] and
[tex]d=\sqrt{17}[/tex]. So AS = √17.
Now we can fill in our area formula:
[tex]A=\frac{1}{2}(\sqrt{17})(3\sqrt{17}+\sqrt{17})[/tex]
Simplifying a bit:
[tex]A=\frac{1}{2}(\sqrt{17})(4\sqrt{17})[/tex] and simplifying a bit more:
[tex]A=\frac{4*17}{2}[/tex] and
A = 34
