Answer:
45
Explanation:
You want the area of triangle RST using the area formula A=1/2bh, given the points R(-2, 7), S(-5, 1), and T(7, -5).
Side lengths
A plot of the points is shown in the first attachment. By counting grid squares, you can see that segment RS has a "rise" of 2 grid squares for each 1 to the right. The total (run, rise) is ...
R -S = (-2, 7) -(-5, 1) = (-2 +5, 7 -1) = (3, 6)
The length of RS is found using the Pythagorean theorem (distance formula). It is ...
RS = √(3² +6²) = √45 = 3√5
Similarly the length of segment ST is ...
T -S = (7, -5) -(-5, 1) = (7 +5, -5 -1) = (12, -6)
ST = √(12² +(-6)²) = √180 = 6√5
Slopes
We note that the slopes of these segments are opposite inverses of each other:
slope RS = 6/3 = 2
slope ST = -6/12 = -1/2
This means the segments are at right angles. One of them can be considered to be the "base" and the other the "height" of the triangle.
Area
Using the area formula, we find the area to be ...
A = 1/2bh
A = 1/2(6√5)(3√5) = (1/2·6·3)(√(5·5)) = 9·5 = 45
The area of ∆RST is 45 square units.
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Alternate solution
When the coordinates of a polygon are given, there are several other ways to find its area. One of these is illustrated in the second attachment.
The method illustrated here computes successive "determinants", then finds the area as half the absolute value of their sum. (The sign of the sum will depend on the order in which the points are listed around the figure. Here, it is counterclockwise.) As you can see, we get the same result. You can also see that a spreadsheet is useful for doing the repetitive math.
Area ∆RST = 45 square units
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Additional comment
The distance formula for the length of the segment between two points is ...
d = √((x2-x1)² +(y2-y1)²)
Above, we calculated the differences (x2-x1, y2-y1) separately, then used the "root sum squares" formula for the distance. This has the advantage that (y2-y1)/(x2-x1) is the slope of the segment, and we needed to make sure the segments were perpendicular.