An isosceles triangle ABC is inscribed in a circle, with AB AC. Show that the altitude from A to BC Class 9

An isosceles triangle ABC is inscribed in a circle, with AB AC. Show that the altitude from A to BC Class 9

Question 1.

An isosceles triangle ABC is inscribed in a circle, with AB AC. Show that the altitude from A to BC passes through the centre of the circle. Class 9

Answer:

Given: An isosceles triangle ABC is inscribed in a circle with AB = AC.

Construction: Draw AD ⊥ BC.

∴ AD is the altitude of ∆ABC.

Proof:

To show: Altitude AD passes through the centre O of the circle.

In ∆ABD and ∆ACD,

AB = AC .......[Given]

AD = AD ... [Common side]

∠ADB = ∠ADC = 90° .....[AD is perpendicular to BC]

∴ ∆ABD ≅ ∆ACD ...[By RHS congruence]

∴ BD = CD ... [Corresponding parts of congruent triangles]

∴ D is the midpoint of BC.

Also, AD is perpendicular to BC.

∴ AD is the perpendicular bisector of chord BC.

The perpendicular bisector of a chord passes through the centre of the circle.

∴ AD passes through the centre O of the circle.