Question 5 of 11intermediate🔍 AnalyzeLong Answer4 marks

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

Correct Answer

Given a circle with centre O and diameter AB. Let ll be the tangent at point A and mm be the tangent at point B. We need to prove that tangent ll is parallel to tangent mm.

According to Theorem 10.1, the tangent at any point of a circle is perpendicular to the radius through the point of contact. Since ll is the tangent at A, it is perpendicular to the radius OA. Therefore,

∠OAL=90∘\angle OAL = 90^\circ

Similarly, since mm is the tangent at B, it is perpendicular to the radius OB. Therefore,

∠OBM=90∘\angle OBM = 90^\circ

We observe that ∠OAL\angle OAL and ∠OBM\angle OBM are equal (90∘90^\circ each). These angles act as alternate interior angles formed by the transversal AB intersecting the lines ll and mm.

Since the alternate interior angles are equal, the lines ll and mm are parallel. Hence, the tangents drawn at the ends of a diameter of a circle are parallel.

∴ The tangents ll and mm are parallel.

Exercise: EXERCISE 10.2 | Q: 4 | (Page 9)
For More Understanding

Explanation

This proof relies directly on Theorem 10.1 provided in the context, which establishes the perpendicular relationship between a tangent and the radius at the point of contact. By applying this theorem to both tangents, we establish that both form 90∘90^\circ angles with the diameter.

This geometric configuration allows the student to apply the property of parallel lines (equal alternate interior angles) to complete the proof.

Solution Steps

  1. Step 1: Identify the given elements: A circle with centre O, diameter AB, tangent ll at A, and tangent mm at B.

  2. Step 2: Apply Theorem 10.1 to establish that tangent ll is perpendicular to radius OA.

    ∠OAL=90∘\angle OAL = 90^\circ

  3. Step 3: Apply Theorem 10.1 to establish that tangent mm is perpendicular to radius OB.

    ∠OBM=90∘\angle OBM = 90^\circ

  4. Step 4: Compare the angles:

    ∠OAL=∠OBM=90∘\angle OAL = \angle OBM = 90^\circ

  5. Step 5: Conclude that since these equal angles are alternate interior angles formed by transversal AB on lines ll and mm, the lines ll and mm are parallel.

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