1. (12 pts) Ia the diagram below, chords AB (12 Pts of the n below, chords AB an

Question

1. (12 pts) Ia the diagram below, chords AB (12 Pts of the n below, chords AB and CD are perpendicular. Let the be given by d. Prove that Ax+BXCx+ DX2 - I diameter of the cirele be gi I will start you off. Construct all of the following: diameter CF, as well as segments AD, BC, and BF. Continue. Hint I: Now what? Well, write out some equations. Look at them. You should notioe two segments that you would LOVE to be congruent. But how to prove? Arc chase! Hint 2: And, as it turns out, Chord Chord Segment Theorem is not going to help you here. But looking at the equation you need to prove, it should be clear what theorem will be helpful.

Explanation / Answer

Here in each right triangle, form ing here we will apply pythagorean theorem,

c^2=a^2+b^2

so in triangle AXD

AX^2+XD^2= AD^2               .......(i)

in triangle CXB,   CB^2=CX^2+XB^2            .........(ii)

now on adding both sides of each equation,w we get

AX^2+ BX^2+ CX^2 + DX^2= AD^2+ CB^2            .........(iii)

Now again in triangle CBF, using the same theorem, we have

d^2= CB^2+ BF^2

But BF= AD   (as both chord are at equal distance from center based on line AB)

so d^2= AD^2+ CB^2 .......(iv)

so by (iii) and (iv), we finally get,

AX^2+ BX^2+ CX^2 + DX^2=d^2

Proved.

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