1. ## Finding radius of circle with 2 Perpendicular Chords

Two chords KV and QR of circle O are perpendicular at point P, with PQ=6 and PR=8 If radius of circle is sq root of 65. find KP and PV.

The answer is KP=12 and PV=4

My ques is how did they get KP? i understand how they get PV since QP*PR=KP*PV

But i dont know what to do with the radius to get KP?

2. ## Re: Finding radius of circle with 2 Perpendicular Chords

Two chords KV and QR of circle O are perpendicular at point P, with PQ=6 and PR=8 If radius of circle is sq root of 65. find KP and PV.

The answer is KP=12 and PV=4

My ques is how did they get KP? i understand how they get PV since QP*PR=KP*PV

But i dont know what to do with the radius to get KP?
1. Draw a sketch.

2. Let x denote $|\overline{VP}|$ and y denotes $|\overline{PK}|$.
According to the intersecting chord theorem (google for it!)[1st equation] and Pythagorean theorem[2nd equation] you'll get the system of equations:

$\left|\begin{array}{rcl}x \cdot y &=& 6 \cdot 8 \\ \left(\dfrac{x+y}{2}\right)^2+1^2 &=& 65 \end{array}\right.$

3. Solve for x and y.

3. ## Re: Finding radius of circle with 2 Perpendicular Chords

I understand the intersecting chord theorem. But how did you derive the 2nd equation?

4. ## Re: Finding radius of circle with 2 Perpendicular Chords

I understand the intersecting chord theorem. But how did you derive the 2nd equation?
The centre lies on the perpendicular bisector of QR,
which is parallel to KV
and lies 1 unit below KV (from the sketch).
Therefore Pythagoras' Theorem applied to the right-angled triangle OKM or OVM,
where O is the centre and M is the midpoint of KV yields Earboth's 2nd equation.

5. ## Re: Finding radius of circle with 2 Perpendicular Chords

Thanks. i get it now. i was making it way to complicated

6. ## Re: Finding radius of circle with 2 Perpendicular Chords

You could also have answered exclusively with Pythagoras' Theorem.

Label the circle centre O, the midpoint of KV = M and the midpoint of QR = N.

Then the circle centreline containing O and N lies 1 unit below [KV]
as it is 7 units from Q and 7 units from R.

M is the midpoint of [KV]

$\Rightarrow\ |KM|^2+1=65\Rightarrow\ |KM|=8$

Label the point of intersection of [QR] and the horizontal centreline as S.

Then

$|OS|^2+7^2=65\Rightarrow\ |OS|=4$

Then

$|KV|=2|KM|,\;\;\;|KP|=|KM|+|OS|,\;\;\;|PV|=|KM|-|OS|$

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# two perpendicular chords of circle given. how to find radius of that circle

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