Question

In the figure, O is the centre of the
circle. Seg AB is the diameter.
Tangent at A intersects the secant
BD at C. Tangent HD intersects
side AC at J then prove :
seg AJ = seg CJ.​

Answer 1

Explanation:

AJ and DJ are the tangents from the same external point J.

Property of tangents of circle:

Two tangents drawn from the same external point are equal in length.

Therefore, AJ = DJ.

OA is radius which is perpendicular to the tangent AJ.

ΔBAC is a right triangle.

Sum of the angles of the triangle = 180°

⇒ ∠BAC + ∠ABC + ∠ACB = 180°

⇒ 90° + ∠ABC + ∠ACB = 180°

⇒ ∠ABC + ∠ACB = 90°

Angle in the semi-circle is always 90°.

⇒ ∠ADB = 90°

Therefore, ΔADB is a right triangle.

Sum of the adjacent angles in a straight line = 180°

⇒ ∠ADB + ∠ADC = 180°

⇒ 90° + ∠ADC = 180°

⇒ ∠ADC = 90°

In ΔJAD, AJ = DJ, so it is isosceles triangle.

Therefore, ∠JAD = ∠JDA

∠JDC + JDA = 90°

∠JDC = JCD

⇒ DJ = JC

But, DJ = AJ

Therefore, AJ = CJ

Hence proved.