Question

Let A9A1 A2 A3 A4 A5 be a regular hexagon inscribed in circle of unit radius. Then
the product of the lengths of the line segments A, A1, A, A2, and AAs is

Answer 1

Step-by-step explanation:

Given,

A unit circle with centre O and a regular hexagon with each of its vertices on the circumference on the circle

A unit circle means a circle with radius 1 unit

Let the Hexagon be ABCDEF because it has 6 sides

Now we can divide a regular hexagon into 6 equilateral triangles

Let's just prove that fast

also let me write angle as an.

an.AOB + an.BOC + an.COD + an.DOE + an.EOF + an.FOA = 360°

we know that they are cut into equal parts so each angle will be same

so,

an.AOB + an.AOB + an.AOB + an.AOB + an.AOB + an.AOB = 360°

6 × an.AOB = 360°

an.AOB = 360 ÷ 6 = 60°

so each angle is same and equal to 60°

OA = OB = OC = OD = OE = OF = 1 unit

Radii of the same circle

so in triangle OAB,

OA = OB

so, an.OAB = an.OBA (property of isosceles triangle)

Now,

an.AOB + an.OAB + an.OBA = 180°

60° + an.OAB + an.OBA = 180°

an.OAB + an.OBA =180 - 60 = 120°

Now, an.OAB = an.OBA

an.OAB + an.OAB = 120

2 × an.OAB = 120

an.OAB = 60°

so, an.AOB = an.OAB = an.OBA = 60°

Hence it is an Equilateral Triangle

Thus, OA = OB = AB

Now OA = 1 unit

so, AB = 1 unit

we also know that AB = BC = CD = DE = EF = AF = 1 unit

so product of the sides will be 1 × 1 × 1 × 1 × 1 × 1 = 1⁶ = 1 unit

Hence the product of the sides of a regular Hexagon inscribed in a circle of radius 1 unit is 1 unit

Hope you understood it........All the best

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