Question

Prove Theorem 8: If A and B are convex then so also is A intersection B

Theorem 8 -- If A and B are convex, then so also is A intersection B

Answer #1

Prove the following:
Claim: Consider a convex quadrilateral □ABCD.
If σ(□ABCD) = 360, then σ(▵ABC) = σ(▵ACD) = 180.
Hint: Use the Split Triangle Theorem and/or the Split
Quadrilateral Theorem
(The angle sum of a convex
quadrilateral is the sum of the angle sum of the two triangles
formed by adding a diagonal.
Recall: The angle sum of quadrilateral □ABCD is σ(□ABCD) =
m∠ABC + m∠BCD +
m∠CDA + m ∠DAB.)

prove that every disc is convex

Prove that the perspective projection of convex polyhedron of 3D
space is a convex polygon on a 2D image plane.

Prove Pythagorean Theorem
a) Show at least 3 different proves if the Pythagorean
Theorem
b) Pythagorean triples with applications: indirect measurements
and mental math problems

Using the pigeonhole theorem prove that an algorithm cannot
losslessly compress any 1024-bit binary string so that it's not
more than 1023 bits.

Prove the IVT theorem
Prove: If f is continuous on [a,b] and f(a),f(b) have different
signs then there is an r ∈ (a,b) such that f(r) = 0.
Using the claims:
f is continuous on [a,b]
there exists a left sequence (a_n) that is increasing and
bounded and converges to r, and left decreasing sequence and
bounded (b_n)=r.
limf(a_n)= r= limf(b_n), and f(r)=0.

At an intersection of hospital hallways, a convex spherical
mirror is mounted high on a wall to help people avoid collisions.
The magnitude of the mirror's radius of curvature is 0.574 m. (a)
Locate the image of a patient 10.6 m from the mirror. (Use the
correct sign conventions.)
cm (from the mirror)
(b) Indicate whether the image is upright or inverted.
upright inverted
(c) Determine the magnification of the image.

Prove that Contractible sets and hence convex sets are
connected.

At an intersection of hospital hallways, a convex spherical
mirror is mounted high on a wall to help people avoid collisions.
The magnitude of the mirror's radius of curvature is 0.542 m.
(a) Locate the image of a patient 10.2 m from the mirror. (Use
the correct sign conventions.)
Apply the mirror equation using the correct focal length and object
distance to find the image distance. cm (from the mirror)
(b) Determine the magnification of the image.

Prove the angle subtraction theorem

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