Question

an isosceles triangle ABC is inscribed in a circle if a b equal to AC equal to 13 centimetre and BC equal to 10 cm find the radius of the circle​

Answer 1

If AB equal to AC equal to 13 cm and BC equal to 10 cm of triangle ABC inscribed in a circle then the radius of the circle is 7 cm.

Step-by-step explanation:

It is given,

ABC is an isosceles triangle inscribed in a circle.

AB = AC = 13 cm

BC = 10 cm

Step 1:

Referring to the figure attached below,

Let's draw AD as the median of the isosceles ∆ ABC such that AD perpendicular to BC.

We know that the median drawn to an isosceles triangle is the perpendicular bisector of the base as well as the angle bisector of the angle opposite to the base.

∴ BD = DC = ½ * BC = ½ * 10 = 5 cm  

Let’s consider the right-angled ∆ ABD and by applying Pythagoras theorem, we get

AD = √[AB² – BD²]

⇒ AD = √[13² – 5²]

⇒ AD = √[144]

⇒ AD = 12 cm

Step 2:

Let the radius of the circle with centre O be “r” cm.

OA = OB = OC = r cm

Since AD = 12 cm ∴ OD = [12 – r] cm

Now, consider the right-angled ∆ OBD and by applying Pythagoras theorem, we get

OB² = OD² + BD²

⇒ r² = [12 - r]² + 5²

⇒ r² = 144 -24r + r² + 25

⇒ 24 r = 169

⇒ r = $\frac{169}{24}$

⇒ r = 7.04 ≈ 7 cm

Thus, the radius of the circle is 7 cm.

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