Let be the set of all students in the <u>c</u>lassroom.
Let and
be the sets of students that pass <u>p</u>hysics and <u>m</u>ath, respectively.
We're given
i. We can split up into subsets of students that pass both physics and math
and those that pass only physics
. These sets are disjoint, so
ii. 9 students fails both subjects, so we find
By the inclusion/exclusion principle,
Using the result from part (i), we have
and so the probability of selecting a student from this set is
9514 1404 393
Answer:
a. (1.4, 2.6)
b. (3, 2)
Step-by-step explanation:
a. The circumcenter is the intersection point of the perpendicular bisectors of the sides of the triangle.
For points X(a, b) and Y(c, d), the perpendicular bisector of XY can be written as ...
(c -a)(x -(c+a)/2) +(d -b)(y -(d+b)/2) = 0
Then the equation of the perpendicular bisector of MN is ...
(-2-4)(x -(-2+4)/2) +(4-0)(y -(4+0)/2) = 0
-6(x -1) +4(y -2) = 0
3x -2y = -1 . . . . divide by -2 and put in standard form
Similarly, the perpendicular bisector of NO is ...
(0-(-2))(x -(0-2)/2) +(6 -4)(y -(6+4)/2) = 0
2(x +1) +2(y -5) = 0
x + y = 4 . . . . divide by 2 and put in standard form
So, the point of intersection is ...
(3x -2y) +2(x +y) = (-1) +2(4) . . . . . add twice the 2nd equation to the 1st
5x = 7 . . . . . . simplify
x = 1.4 . . . . . . divide by 5
y = 4-1.4 = 2.6 . . . . . find y from the second equation
The coordinates of the circumcenter are (1.4, 2.6).
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b. The centroid is the average of the vertex coordinates.
centroid = (A +B +C)/3 = ((1, 2) +(3, 4) +(5, 0))/3 = (1+3+5, 2+4+0)/3 = (9, 6)/3
centroid = (3, 2)
