circle geometry year 11 extension 1 questions please help! (1 Viewer)

abdullion

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So I've been struggling to answer these two questions, please try help if you can!

1. CD is a diameter of the circle. CE, OF and DG are each perpendicular to the line EXFYG.(X and Y are points on the circle, and EG cuts the circle at two points.) Prove that EX = YG.

2. PAB is an isosceles triangle inscribes in a circle of radius 26mm. If PA = PB and AB = 48mm, calculate the length of AP.
 

integral95

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I don't really understand the first one lol.


For the 2nd one

Let the Midpoint of AB be M

construct OM (where O is the center) and you'll see that angle OMA is a right angle as a radius bisects a chord at right angles.

From triangle OMA, use Pythagoras to find OM (i got 10 mm), you'll see that MD = OD + OM = 10+ 26 = 36

then use Pythagoras in triangle AMD to find AD (AM = 24, MD = 36) etc etc.
 

Feynman

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For 1:

FX = YF (line from the centre that meets a chord at right angles bisects it)

CO = EX + FX (opposite sides in a rectangle are equal)
Then CO - FX = EX
and DO - YF = YG (similarly)
But CO = DO (radii of a circle are equal)
And FX = YF (proven above)

Therefore CO - FX = EX and CO - FX = YG
And therefore, EX = YG
 
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InteGrand

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So I've been struggling to answer these two questions, please try help if you can!

1. CD is a diameter of the circle. CE, OF and DG are each perpendicular to the line EXFYG.(X and Y are points on the circle, and EG cuts the circle at two points.) Prove that EX = YG.












See if you can prove Case 2, where X and Y are on opposite sides of CD.
 
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InteGrand

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So I've been struggling to answer these two questions, please try help if you can!

1. CD is a diameter of the circle. CE, OF and DG are each perpendicular to the line EXFYG.(X and Y are points on the circle, and EG cuts the circle at two points.) Prove that EX = YG.


 

InteGrand

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For 1:

FX = YF (line from the centre that meets a chord at right angles bisects it)

CO = EX + FX (opposite sides in a rectangle are equal)
Then CO - FX = EX
and DO - YF = YG (similarly)
But CO = DO (radii of a circle are equal)
And FX = YF (proven above)

Therefore CO - FX = EX and CO - FX = YG
And therefore, EX = YG
COFX need not be a rectangle. In the case that it is, your proof is valid.
 

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