Question

Seg AB is a diameter of a circle with Centre P. Seg AC is a chord. A secant through point P and parallel to seg AC intersects the tangent drawn at C in D. Prove that line DB ia s tangent to the circle. ​

Answer 1

Answer:

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Answer 2

Proved that line DB ia s tangent to the circle. ​

To prove : Line DB is a tangent to the circle.

Given :

• Segment AB is a diameter of a circle with center P.
• Segment AC is a chord.    
• A secant through point P and parallel to seg AC intersects the tangent drawn at C in D.

Where, a tangent CD is drawn at point C.

If the line is perpendicular to the radius drawn to the point, the line is a tangent to a circle.

(i.e.,) CP ⊥ CD

∠PCD = 90°    -----> ( 1 )

Take ΔAPC,

Radius of the circle,

AP = CP

Where, angles opposite to equal sides are equal.

∠PAC = ∠PCA   -----> ( 2 )

Here, AC ║ PD

∠PCA = ∠CPD   [ They are alternate angles ]  -----> ( 3 )

∠PAC = ∠DPB   [ They are corresponding angles ] -----> ( 4 )

From the equations ( 1 ), ( 2 ) and ( 3 ), we get

∠CPD = ∠DPB   -----> ( 5 )

Now, in ΔCPD and ΔBPD, we have

PD = PD   [ Both triangles have common sides ]

From equation ( 5 ),

Radius of the circle,

∠CPD = ∠DPB

Hence, by SAS test

Δ CPD ≅ Δ BPD

By corresponding parts of congruent triangles,

∠PCD = ∠PBD   -----> ( 6 )

From equation ( 1 )  and ( 6 ), we get

∠PBD = 90°

PB ⊥ DB

By the converse tangent theorem :

The line is tangent to the circle, when a line is perpendicular to the radius at its endpoint then the line is tangent to the circle.

Hence, proved that DB is tangent to the circle at point B.

To learn more...

1. Seg AB is a diameter of a circle with centre p. Segment AC is a chord .A secant Through p and parallel to seg AC intersects the tangent drawn at C in D prove that line DB is a tangent to the circle

brainly.in/question/14336382

2. In the figure, O is the centre of the circle, Seg AB is diameter. At the pt C on the circle the tangent CD is drawn. Line BD is also a tangent to the circle at pt B .Show that seg OD || chord AC​

brainly.in/question/8727278