Pj Problems - Overview
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The Human-Body Systems
Faith, Love, Charity
The right triangle XYZ of figure 121.1 has sides of lengths x and y, and hypotenuse of length z. Its area is z2/4. Using the Forward-Backward proof techniques, proof that triangle XYZ is isosceles.
S7P2A21 (Identity - Physical Property).
Pj Problem of Interest is of type identity (physical property). Proofs establish truths. So they are identity problems.
The Forward-Backward proof techniques are generally used to establish the If A then B truth. Where A and B are statements. The statements in the given problem are:
A: Right triangle XYZ with sides of length x and y, and hypotenuse of length z has an area of x2/4.
B: Right triangle XYZ is isosceles.
Proof of Interest: If A then B (A implies B).
Forward Process Method:
Proof begins with assuming A is true then progresses stepwise with intermediary truths given the truth of A until B is established.
Truth of A:
Area =xy/2 = z2/4 ------(1).
x2 + y2 = z2 (Pythagoras Theorem)-------(2)
So, xy/2 = (x2 + y2)/4
So, 0 = (x -y)2
So, (x - y) = 0
So, x = y
So, B is established.
So If A then B (A implies B).
Backward Process Method:
Proof begins with assuming B is true then progresses stepwise with intermediary truths given the truth of B until A is established.
x = y.
So, (x -y)2 = 0.
So, x2 -2xy + y2 = 0.
So, z2 = 2xy
So, z2/4 = xy/2 = area of XYZ
So, A is established.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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