AB is the diameter of the circle with centre O. From a point P on the circle, a perpendicular PN is drawn on AB. Prove geometrically that PB² = AB.BN.
WBBSE
Class X
Mathematics
Theorem Related To Angle In A Circle
05 Mark
2025
Given: AB is the diameter of a circle with center O. From a point P on the circle, a perpendicular PN is drawn on AB.
To Prove: PB2 = AB × BN
Proof: AB is the diameter, so ∠ APB = 90º
PN is perpendicular to AB.
Triangles PNB and APB are similar by AA similarity.
By similarity property,
\frac{PB}{AB} = \frac{BN}{PB}
Cross multiplying,
PB2 = AB × BN (Hence proved)
Related Questions
More WBBSE Moderate level questions
1/12
Q.
Prove that in any circle, angles in the same segment are equal.
A.
Given: Let ∠ACB and ∠ADB be any two angles in the segment...
Q.
AOB is a diameter of a circle whose center is O. The point C lies...
A.
Given: AOB is a diameter of the circle with center O. Point...
Q.
State True/False: “The equal chords of a circle are at equal distances from the centre...
A.
True Explanation: In a circle, chords that are equal in length are...
Q.
O is the circumcentre of the triangle ABC. OD ⊥ BC. Prove that ∠BOD =...
A.
Given: O is the circumcentre of triangle ABC. OD is perpendicular to...
Q.
AB is a diameter of a circle with centre O. If the chord AC subtends...
A.
(b) 30° Explanation: In a circle, the angle subtended by a chord...
Q.
The angle in the segment of a circle which is greater than a semicircle is...
A.
The angle in the segment of a circle which is greater than...
Q.
Fill up the blank : The front angle formed at the center of a circle...
A.
The front angle formed at the center of a circle by an...
Q.
What is the value of a semicircular angle?
A.
90° Explanation: An angle standing on a semicircle (i.e., subtended by a...
Q.
‘O’ is the center of a circle and PQ is a diameter. R is a...
A.
(d) 45° Explanation: Since PR = RQ, triangle PRQ is isosceles, and...
Q.
Distance between two pillars is 150 m, height of one pillar is three times the...
A.
Let the height of the smaller pillar be h meters Then, the...
Q.
From the middle of a ground Habu observes a flying bird toward north at an...
A.
Let the initial position of the bird be A and the final...
Q.
Draw a triangle whose sides are of lengths 6 cm, 8 cm, and 10 cm....