Question

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.

Answer #1

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 = (1,4), B = (0,−9), C = (7,2), and D = (6,9). Prove that
angles DAB and DCB are the same size. Can anything be said about
the angles ABC and ADC?

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?

Prove
a)p→q, r→s⊢p∨r→q∨s
b)(p ∨ (q → p)) ∧ q ⊢ p

Give the indirect proofs of:
p→q,¬r→¬q,¬r⇒¬p.p→q,¬r→¬q,¬r⇒¬p.
p→¬q,¬r→q,p⇒r.p→¬q,¬r→q,p⇒r.
a∨b,c∧d,a→¬c⇒b.

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q.
I need to prove that:
a) R is an equivalence relation. (which I have)
b) The equivalence classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq
I...

Let A=(0,0), B=(1,1), C=(-1,1), A'(2,0),B'(4,0), C'(2,-2).Show
that triangle ABC and triangle A'B'C' satisfy the hypothesis of
Proposition 2.3.4 in taxicab geometry but are not congruent in
it

1) Let a = 〈4,−5,−2〉 and b = 〈2,−4,−5〉 Find the projection of b
onto a. proj a b=
2) Find the area of a triangle PQR, where P=(−2,−4,0),
Q=(1,2,−1), and R=(−3,−6,5)
3) Complete the parametric equations of the line through the
points (7,6,-1) and (-4,4,8)
x(t)=7−11
y(t)=
z(t)=

Let A = (2, 9), B = (6, 2), and C = (10, 10). Verify that
segments AB and AC have the same length. Measure angles ABC and
ACB. On the basis of your work, propose a general statement that
applies to any triangle that has two sides of equal length. Prove
your assertion, which might be called the Isosceles-Triangle
Theorem.

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Prove that R is an equivalence
relation.
you may use the following lemma: If p is prime and p|mn, then
p|m or p|n

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