Question

Let A = (0,0), B = (3,0), C = (2,8). Find a point P such that AP is perpendicular BC, BP is perpendicular to AC, and CP is perpendicular to AB. Does it surprise you that such a point exists?

Answer #1

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Consider the triangle with vertices (0,0),(3,0),(0,3) and let P
be a point chosen uniformly at random inside the triangle. Let X be
the distance from P to (0,0). (i) Is X a random variable? Explain
why or why not. (ii) Compute P(X≥0), P(X≥1), P(X≥2), and
P(X≤3).

. Let A = (−2, 4) and B = (7, 6). Find the point P on the line y
= 2 that makes the total distance AP + BP as small as possible.

Consider a cicle with AB as diameter and P another point on the
circle. Let M be the foot of the perpendicular from P to AB. Draw
the circles which have AM and M B respectively as diameters, which
meet AP at Q are BP are R. Prove that QR is tangent to both
circles.
Hint: As well as the line QR, draw in the line segments M Q and
M R.

Let P[A] = P[B] = 1/3 and P[A ∩ B] = 1/10.
Find the following:
(a) P[A ∪ Bc ]
(b) P[Ac ∩ B]
(c) P[Ac ∪ Bc ]
Now, let P[A] = 0.38 and P[A ∪ B] = 0.84.
(d) For what value of P[B] are A and B mutually exclusive?
(e) For what value of P[B] are A and B independent?

ABC is a right-angled triangle with right angle at A, and AB
> AC. Let D be the midpoint of the side BC, and let L be the
bisector of the right angle at A. Draw a perpendicular line to BC
at D, which meets the line L at point E. Prove that
(a) AD=DE; and
(b) ∠DAE=1/2(∠C−∠B)
Hint: Draw a line from A perpendicular to BC, which meets BC in
the point F

Let A and B be independent events of some sample space. Using
the definition of independence P(AB) = P(A)P(B), prove that the
following events are also independent:
(a) A and Bc
(b) Ac and B
(c) Ac and Bc

Let ABCD be a rectangle with AB = 4 and BC = 1. Denote by M the
midpoint of line segment AD and by P the leg of the perpendicular
from B onto CM.
a) Find the lengths of P B and PM.
b) Find the area of ABPM.
c) Consider now ABCD being a parallelogram. Denote by M the
midpoint of side AD and by P the leg of the perpendicular from B
onto CM. Prove that AP =...

Let A = (0,0), B = (8,1), C = (5,−5), P = (0,3), Q = (7,7), and
R = (1,10). Prove that angles ABC and PQR have the same size.

Let A and B represent events such that P(A) = 0.6, P(B) = 0.4,
and P(A ∪ B) = 0.76. Compute: (a) P(A ∩ B) (b) P(Ac ∪ B) (c) P(A ∩
Bc ) (d) Are events A and B mutually exclusive? Are they
independent? Explain by citing the definitions of mutual
exclusivity and independence.

In the rectangle ABCD, AB = 6 and BC = 8. The diagonals AC and
BD intersect at O. Point P lies on the diagonal AC such that AP =
1. A line is drawn from B through P and meets AD at S. Let be R a
point on AD such that OR is parallel to BS. a) Find the lengths of
AS and RD. Hint: Denote AS = x. Use P S k OR and OR k BS...

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