$A B$ and $A C$ are two chords of a circle of radius $r$ such that $A B = 2 A C$ . If $p$ and $q$ are the distances of $A B$ and $A C$ from the centre, then show that $4 q^{2} = p^{2} + 3 r^{2}$ .

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Solution

Consider the diagram shown below.

Given:

$A B$ and $A C$ are two chords of a circle with center $O$ . Such that $A B = 2 A C$

$p$ and $q$ are $⊥$ distances of $A B$ and $A C$ from center $O$ i.e., $O M = p$ and $O N = q$

$r$ is the radius of the circle

To prove that:

$4 q^{2} = p^{2} + 3 r^{2}$

Proof:

Join $O A$ .

$O M$ and $O N$ are $⊥$ distances of $A B$ and $A C$ from center $O$ .

Here,

$A N = \frac{A C}{2}$ (perpendicular from center to chord intersect at mid-point of the chord)

$A M = \frac{A B}{2}$ (perpendicular from center to chord intersect at mid-point of the chord)

In right angled $Δ O M A$ ,

$O M^{2} + A M^{2} = O A^{2}$

$p^{2} + A M^{2} = r^{2}$

$A M^{2} = r^{2} - p^{2}$ …… (1)

In right angled $Δ O N A$ ,

$O N^{2} + A N^{2} = O A^{2}$

$q^{2} + A N^{2} = r^{2}$

$A N^{2} = r^{2} - q^{2}$ …… (2)

Since,

$A M = \frac{A B}{2} = \frac{2 A C}{2} = A C = 2 A N$

From equations (1) and (2), we have

$r^{2} - p^{2} = A M^{2}$

$r^{2} - p^{2} = 4 A N^{2}$

$r^{2} - p^{2} = 4 [r^{2} - q^{2}]$

$4 q^{2} = p^{2} + 3 r^{2}$

$L H S = R H S$

Hence, proved.

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Similar questions

AB and AC are two chords of a circle of radius $r$ such that $A B = 2 A C$ . If $p$ and $q$ are the distances of $A B$ and $A C$ from the centre then prove that $4 q^{2} = 3 r^{2} + p^{2}$

Q. Question 12
AB and AC are two chords of a circle of radius r such that AB = 2AC. If p an q are the distances of AB an AC from the centre, prove that

4 q^{2} = p^{2} + 3 r^{2}

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4 q^{2} = p^{2} + 3 r^{2}

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AB and AC are two chords of a circle of radius r such that AB=2AC. If p and q are the distances of AB and AC from the centre, then show that 4q2=p2+3r2.

$A B$ and $A C$ are two chords of a circle of radius $r$ such that $A B = 2 A C$ . If $p$ and $q$ are the distances of $A B$ and $A C$ from the centre, then show that $4 q^{2} = p^{2} + 3 r^{2}$ .