Ex.10.2 Q9 Circles Solution - NCERT Maths Class 10

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Question

In Figure, \(XY\) and \(X'Y' \) are two parallel tangents to a circle with center \(O\) and another tangent \(AB\) with point of contact \(C\) intersecting \(XY\) at \(A\) and \(X'Y' \) at \(B.\)

Prove that \(\angle {AOB} = {90^ \circ }\).

Text Solution

What is known?

  •  '\(O\)' is the centre of the circle.
  • \(XY\) and \(X'Y'\)' are the two parallel tangents to the circle.
  • \(AB\) is another tangent with point of contact \(C\), intersecting \(XY\) at \(A\) and \(X'Y'\) at \(B.\)

To prove:

\(\angle {AOB} = {90^ \circ }\)

Reasoning :

Join point \(O\) to \(C.\)

In \(\Delta OPA\) and \(\Delta OCA\)

\(OP = OC\) (Radii of the circle are equal)

\(AP = AC \) (The lengths of tangents drawn from an external point \(A\) to a circle are equal.)

\(AO = AO\)  (Common)

By \(SSS\) congruency, \(\Delta {OPA} \cong \Delta {OCA}\)

SSS Congruence Rule: If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

Steps:

Therefore, \(\angle {OPA} = \angle {OCA}\)

Similarly, \(\Delta {OCB} \cong \Delta {OBQ}\)

Therefore, \(\angle {COB} = \angle {BOQ}\)

\(PQ\) is a diameter, hence a straight line and \(\angle {POQ} = {180^ \circ}\)

But \(\begin{align} \angle {P O Q} = \angle {P O A} + \angle {A O C} + \angle {C O B} + \angle {B O Q} \end{align}\)

\[\begin{align} \therefore \quad \angle {POA} + \angle {AOC }+ \angle {C O B} + \angle {B O Q} & = 180 ^ { \circ } \\ 2 \angle {A O C} + 2 \angle {C O B} &= 180 ^ { \circ }\\ (\therefore \quad \angle POA=\angle AOC \;{\rm {and}}\; \angle COB=\angle BOQ) \\ \therefore \angle A O C + \angle C O B & = 90 ^ { \circ } \end{align}\]

From the figure, \(\angle {A O C} + \angle {C O B} = \angle {A O B}\)

\(\therefore \quad \angle {A O B} = 90 ^ { \circ }\)

Hence Proved

  
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