Home
Class 10
MATHS
Seg AB is a diameter of a circle with ce...

Seg AB is a diameter of a circle with centre P. Seg 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.

Text Solution

Verified by Experts

In `Delta PAC` ,
seg `PA cong seg PC " "` …(Radii of the same circle )
` :. Angle PAC cong angle PCA " "` …(Isosceles Delta theorem ) …(1)
seg AC `||` seg PD nd PA is the transversal ,
` angle PAC cong angle BPD " " ` ...(Corresponding angles ) ... (2)
seg AC `||` seg PD and PC is the transversal ,
` angle PCA cong angle CPD " "` ...(ALternate angles ) ...(3)
` :. angle BPD cong angle CPD " "` ... [ From (1) , (2) and (3) ] ...(4)
In `Delta BPD and Delta CPD,`
seg `BP cong seg CP" "` ...(Radii of the same circle )
` angle BPD cong angle CPD " " ` ...[ From (4)]
seg `PD cong seg PD " "` ...(Common side )
` :. Delta BPD cong Delta CPD " "` (SAS test)
` :. angle PBD cong angle PCD " " ` ... (c.a.c.t.)
` angle PCD = 90^(@) " "` ... (Tangent is perpendicular to radius )
` :. angle PBD = 90^(@)`
` :. ` line DB is tangent to the circle at point B
... ( Converse of tangent theorem )
Promotional Banner

Topper's Solved these Questions

  • CHALLENGING QUESTIONS

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise SECTION 3 (MODEL QUESTION PAPER FOR PRACTICE )|7 Videos

  • CHALLENGING QUESTIONS

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise SECTION 3 (MODEL QUESTION PAPER FOR PRACTICE ) Choose the correct alternative :|3 Videos

  • CHALLENGING QUESTIONS

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise EXAMPLE FOR PRACTICE|22 Videos

  • CIRCLE

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Challenging Questions|10 Videos

Similar Questions

Explore conceptually related problems

In the adjoining figure, seg AB is a diameter of a circle with centre O. the bisector of angleACB intersects the circle at point D. Prove that ,seg ADcongseg BD . Complete the following proof by filling the blanks.

In figure, seg AB is a diameter of a circle with centre O. The bisector of /_ACB intersects the circle at point D. Prove that seg AD cong seg BD. Complete the following proof by filling in the blanks

Two circles whose centres are O and O' intersect at P. Through P, a line l parallel to OO' intersecting the circles at C and D is drawn.Prove that CD=2OO

AB is a diameter of a circle and AC is its chord such that angleBAC=30^(@) . If the tengent at C intersects AB extended at D, then BC=BD.

Prove that the tangents drawn at the ends of the diameter of a circle are parallel.

In the figure, seg AB is a diameter of a circle with centre C. Line PQ is a tangent, which touches the circle at point T. seg AP bot line PQ and seg BQ bot line PQ. Prove that, seg CP cong seg CQ.

In the figure, seg AB is a diameter of a circle with centre C. Line PQ is a tangent , which touches the circle at point T. seg AP _|_ line PQ and seg BQ _|_ line PQ. Prove that, seg CP ~= seg CQ.

In the above figure, seg AB is a diameter of a circle with centre P.C is any point on the circle. Seg CE _|_ seg AB. Prove that CE is the geometric mean of AE and EB. Write the proof with the help of folloing steps: (a) Draw ray CE. It intersects the circle at D. (b) Show that CE = ED. Write the result using theorem of intersection of chords insides a circle. (d) Using CE = ED, complete the proof.

Point O is the centre of a circle . Line a and line b are parallel tangents to the circle at P and Q. Prove that segment PQ is a diameter of the circle.

NAVNEET PUBLICATION - MAHARASHTRA BOARD-CHALLENGING QUESTIONS-SECTION 3 (MODEL QUESTION PAPER FOR PRACTICE ) Solve any one of the following subquestions :
  1. Seg AB is a diameter of a circle with centre P. Seg AC is a chord. A s...

    Text Solution

    |

  2. Solve any one of the following subquestions : Draw a circle with c...

    Text Solution

    |