Answer:
8. yes
9. no
10. x = 9
11. x = 2
Explanation:
A point will lie on the perpendicular bisector of a segment if it is equidistant from the end points. The differences of coordinates between S and each of Q and R are sufficient to tell you if their sum of squares is the same. (The distance is the root of the sum of squares of the coordinate differences.)
There are other ways to determine the desired answer, but they generally involve more math. Of course, a geometry program can tell you immediately if S is on the perpendicular bisector of QR.
8.
S -Q = (4 -(-5), -2 -(-1)) = (9, -1)
S -R = (4 -3, -2 -7) = (1, -9)
The sums of squares of these differences are 9² +1² = 82. The distance from S to each of Q and R is √82, so S lies on the perpendicular bisector.
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9.
S -Q = (-2 -(-5), -5 -4) = (3, -9) . . . . distance = √90
S -R = (-2 -8, -5 -(-3)) = (-10, -2) . . . . distance = √104
The two distances are not the same, so S does not lie on the perpendicular bisector.
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10.
The marked segments are equal length, so ...
14x -37 = 10x -1
4x = 36 . . . . . . . . . add 37-10x
x = 9 . . . . . . . divide by 4
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11.
The marked angles are congruent, so ...
16x -7 = 6x +13
10x = 20 . . . . . . . . add 7-6x
x = 2 . . . . . . . . divide by 10