Question

explain tangent Secant segment theorem with proof step by step


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Answer 1

The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.

For two lines AD and BC that intersect each other in P and some circle in A and Drespective B and C the following equation holds:

|PA| • |PD| = |PB| • |PC|

The theorem follows directly from the fact, that the triangles PAC and PBD are similar. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above.

Just understand. Move one of the secants (example-PD) so that it becomes a tangent.

So just change the above values. By changing values we get,

|PD| • |PD| = |PB| • |PC|

=> |PD²| = |PB| • |PC|

This way a 10th class student can understand.

There is another way if you want you can look at this also:—

Statement:

According to tangent-secant theorem:

"When a tangent and a secant are drawn from one single external point to a circle, square of the length of tangent segment must be equal to the product of lengths of whole secant segment and the exterior portion of secant segment."

This theorem is explained through the following diagram:

Here, a tangent PS and a secant PR are drawn to a given circle from same exterior point P. Then, square of measure of tangent PS is equal to the product of length of whole secant PR and length of exterior secant segment PQ. i.e.

PS2 = PR x PQ

or

(Tangent)2 = Whole Secant x external secant

Proof

In order to prove this theorem, we are going to join QS and RS.

In △PSQ and △PRS

∠PSQ = ∠PRS ______ (By tangent-chord theorem, according to which an angle formed by a tangent and a chord is equal to the angle formed by that chord at another point at the circle.)

angleQPS = angleRPS______ (Common angle)

Thus, △PSQ∼△PRS

$\sf \: PS•PR = PQ•PS$

PS² = PR×PQ

Hence proved.

Answer 2

Explanation:

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The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.

For two lines AD and BC that intersect each other in P and some circle in A and Drespective B and C the following equation holds:

|PA| • |PD| = |PB| • |PC|

The theorem follows directly from the fact, that the triangles PAC and PBD are similar. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above.

Just understand. Move one of the secants (example-PD) so that it becomes a tangent.

So just change the above values. By changing values we get,

|PD| • |PD| = |PB| • |PC|

=> |PD²| = |PB| • |PC|

This way a 10th class student can understand.

There is another way if you want you can look at this also:—

Statement:

According to tangent-secant theorem:

"When a tangent and a secant are drawn from one single external point to a circle, square of the length of tangent segment must be equal to the product of lengths of whole secant segment and the exterior portion of secant segment."

This theorem is explained through the following diagram:

Here, a tangent PS and a secant PR are drawn to a given circle from same exterior point P. Then, square of measure of tangent PS is equal to the product of length of whole secant PR and length of exterior secant segment PQ. i.e.

PS2 = PR x PQ

or

(Tangent)2 = Whole Secant x external secant

Proof

In order to prove this theorem, we are going to join QS and RS.

In △PSQ and △PRS

∠PSQ = ∠PRS ______ (By tangent-chord theorem, according to which an angle formed by a tangent and a chord is equal to the angle formed by that chord at another point at the circle.)

angleQPS = angleRPS______ (Common angle)

Thus, △PSQ∼△PRS

\sf \: PS•PR = PQ•PSPS•PR=PQ•PS

PS² = PR×PQ

______________________

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