Question

Abc is an equilateral triangle inscribed in a circle whose centre is o. What fraction of area

Answer 1

Answer:

Let us consider the figure for this question. ABC is the equilateral triangle inscribed in a circle with center at O and radius r. The square is inscribed inside the triangle ABC. Let x be the side of the square.

Given:

OA =OB = OC = r

AB = BC = CA (Equilateral triangle)

A=B=C=60∘A=B=C=60∘

In △△ ABC,

OA = OB = r

angle OBA=angle OAB=30∘=pangle OBA=angle OAB=30∘=p

(OA, OB and OC bisects the angle at the respective vertices of the equilateral triangle)

So p+p+q=180∘p+p+q=180∘

q=120∘q=120∘

Using Cosine law,

AB2=OA2+OB2−2OA.OBcos(q)AB2=OA2+OB2−2OA.OBcos(q)

AB2=r2+r2−2r2cos(120∘)AB2=r2+r2−2r2cos(120∘)

AB2=2r2−2r2×−12AB2=2r2−2r2×−12

AB2=3r2AB2=3r2

AB=r3–√AB=r3

We know that the height 'AD' of equilateral triangle is given by :

AD=(side)3√2AD=(side)32

AD=(r3–√)×3√2AD=(r3)×32

AD=3r2AD=3r2

Also, from the figure,

OD = AD - AO

OD=3r2−rOD=3r2−r

OD=r2OD=r2

Side of square = 2 x OD = r

Hence, the square with largest area that can be inscribed in the equilateral triangle is

Answer 2

Answer:

Let us consider the figure for this question. ABC is the equilateral triangle inscribed in a circle with center at O and radius r. The square is inscribed inside the triangle ABC. Let x be the side of the square.

Given:

OA =OB = OC = r

AB = BC = CA (Equilateral triangle)

A=B=C=60∘A=B=C=60∘

In △△ ABC,

OA = OB = r

angle OBA=angle OAB=30∘=pangle OBA=angle OAB=30∘=p

(OA, OB and OC bisects the angle at the respective vertices of the equilateral triangle)

So p+p+q=180∘p+p+q=180∘

q=120∘q=120∘

Using Cosine law,

AB2=OA2+OB2−2OA.OBcos(q)AB2=OA2+OB2−2OA.OBcos(q)

AB2=r2+r2−2r2cos(120∘)AB2=r2+r2−2r2cos(120∘)

AB2=2r2−2r2×−12AB2=2r2−2r2×−12

AB2=3r2AB2=3r2

AB=r3–√AB=r3

We know that the height 'AD' of equilateral triangle is given by :

AD=(side)3√2AD=(side)32

AD=(r3–√)×3√2AD=(r3)×32

AD=3r2AD=3r2

Also, from the figure,

OD = AD - AO

OD=3r2−rOD=3r2−r

OD=r2OD=r2

Side of square = 2 x OD = r

Hence, the square with largest area that can be inscribed in the equilateral triangle is