Question

An equilateral triangle is inscribed in a circle if the radius of the circle is 5√3 cm find the side of the triangle​

Answer 1

Given:

• Radius of circle = 5√3cm

Find:

• Side of equilateral triangle.

Diagram:

(see in attachment)

Solution:

In ∆ABC

OB is perpendicular on AC

→ B is the mid point of AC and OA is the bisector of ∠DAC

➜ ∠DAC = 60°.....[Each angle of equilateral triangle is 60°]

➜ ∠OAB = 1/2∠DAC

➜ ∠OAB = 1/2×60

➜ ∠OAB = 30°

Now, In ∆OAB

➣ B/H = cos 30°

➣ AB/AO = cos 30°

➣ AB/5√3 = √3/2

➣ 2AB =5√3(√3)

➣ 2AB = 5×3

➣ 2AB = 15

➣ AB = 15/2cm

Now,

↬AC = 2AB

↬AC = 2×15/2

↬AC = 15cm

Hence, side of equilateral triangle is 15cm

Answer 2

Question -

★ An equilateral triangle is inscribed in a circle if the radius of the circle is 5√3 cm find the side of the triangle.

Given that -

★ Radius of circle is 5√3 cm.

To find -

★ Side of the triangle.

Solution -

★ Side of the triangle = 15 cm (XZ)

Diagram of this question -

★ Kindly see from attachment.

◕ Point Y is the middle point of XZ.

◕ As we can see in the attachment that line XY is perpendicular ( || ) to line XZ of the triangle ( ∆ ) and circle ( O )

◕ XO or OX, bisect at angle ( ∠ ) AXZ.

Knowledge required -

★ The sum of interior angles of an equilateral triangle is always 60°

★ Base / Height = cos30°

★ Value of cos30° = √3/2

Full solution -

~ Let us find the measure of ∠OXY

❥ ∠OXY = ½ × ∠ACZ

❥ ∠OXY = ½ × 60°

❥ ∠OXY = 30°

~ In ∆OXY, let's see what to do..!

❥ Base / Height = cos30°

❥ XY / XO = cos30°

Here,

◕ XY = ?

◕ XO = 5√3 cm

◕ Value of cos30° = √3/2

❥ XY / 5√3 = √3/2

• (÷ = ×) ; (× = ÷)

❥ 2XY = 5√3(√3)

• Let us cancel or multiply

❥ 2XY = 5 × 3

❥ 2XY = 15

❥ XY = 15/2

❥ XY = 7.5 cm

~ Now in side XZ

❥ XZ = 2XY

❥ XZ = 2(7.5)

❥ XZ = 2 × 7.5

❥ XZ = 15.0 cm

❥ XZ = 15 cm

• Henceforth, 15 cm is the side of the triangle.