A day full of math games & activities. Find one near you.

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc

Solution:

Given, a tangent is drawn at the midpoint of an arc of a circle.

We have to prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

From the figure,

Centre of the circle is O.

P is the midpoint of arc APB

PT is a tangent to a circle at P

We know that the radius of the circle is perpendicular to the tangent at the point of contact.

So, OP ⟂ PT

∠OPT = 90°

Since P is the midpoint of arc APB,

arc AP = arc BP

∠AOP = ∠BOP

∠AOM = ∠BOM

Considering triangles AOM and BOM,

OA = OB = radius of circle

OM = OM = common side

Also, ∠AOM = ∠BOM

By SAS criterion, the triangles AOM and BOM are similar.

Corresponding parts of congruent triangles or cpct is used to denote the relation between the sides and the angles of two congruent triangles.

By cpct,

∠AMO = ∠BMO —------------------ (1)

The linear pair angles ∠AMO + ∠BMO = 180°

From (1), ∠AMO + ∠AMO = 180°

2∠AMO = 180°

∠AMO = 90°

So, ∠AMO = ∠BMO = 90°

We know that the corresponding angles are equal.

From the figure,

∠BMO = ∠OPT = 90°

This implies AB || PT

Therefore, the tangent at midpoint of the arc is parallel to the chord joining the end points of the arc.

✦ Try This: Prove that the line joining the midpoint of a chord to the centre of the circle passes through the midpoint of the corresponding minor arc.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10

NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 9

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc

Summary:

It is proven that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc

☛ Related Questions: