Name: Angles of the Semicircle: Right Angle
Uploaded: 2019-07-19
Duration: 4 min 42 s
Description: Watch this video to learn why the angle made by the two ends of diameter with any other point on the circumference of a circle is always 90 degrees.

Question

If two chords of a circle intersect internally or externally, then prove that the product of the lengths of segments are equal.

Accepted Answer

Given,

Two chords $AB$ and $CD$ of a circle intersecting at $P$ internally or externally.

Construction Join $AC$ and $BD$ .
Case I

When $AB$ and $CD$ intersect internally as shown in Figure $(i)$
In $△ ACP$ and $△ DBP$ ,
$∠ APC = ∠ BPD$ [vertically opposite angles]
$∠ C A P = ∠ P D B$ [angles in same segment]

$∠ A C P ~ ∠ DBP$ [by $AA$ similarity criterion]

$\Rightarrow \frac{PA}{PD} = \frac{PC}{PB}$ [ $∵$ corresponding sides of similar triangles are proportional]
$\Rightarrow PA \times PB = PC \times PD$

Case II

When $AB$ and $CD$ intersect externally as shown in Figure $(ii)$ .
In $△ ACP$ and $△ DBP, ∠ CAB = ∠ BDP$ [exterior angle of a cyclic quadrilateral $=$ interior opposite angle]
$∠ P = ∠ P$ [common angle]
$∴ △ ACP ~ △ DBP$ [by $AA$ similarity criterion]
$\Rightarrow \frac{PA}{PD} = \frac{PC}{PB}$ [ $∵$ corresponding sides of similar triangles are proportional]
$\Rightarrow PA \times PB = PC \times PD$
Thus, in both the cases,
$PA \times PB = PC \times PD$

Hence, proved.

If two chords of a circle intersect internally or externally, then prove that the product of the lengths of segments are equal.

Important Questions on Circles

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