Question

a circle is inscribed in a square, an equilateral triangle of side 4√3 cm is inscribed in that circle. the length of the diagonal of the square is??

Answer 1

Given :

A circle is inscribed in a square,

An equilateral triangle is inscribed in that circle.

Side of triangle = 4√3cm.

To find :

Length of diagonal of square.

Solution :

We know that side of an equilateral triangle = 4√3 cm.

This triangle is inscribed in a circle, it means this circle has a Circumradius.

The formula to find the Circumradius :

$= \frac{ \textbf{side \: of \: triangle}}{ \sqrt{3} }$

So,

Circumradius r :

$= \frac{4 \sqrt{3} }{ \sqrt{3} } = 4 \: cm$

As we know that, circle is inscribed in a square,

so now we got the half length of side of that square in the form of Circumradius r.

so,

Side of square, s = 2r

$s = 2 \times 4 = 8 \: cm$

so,

side of square = 8 cm.

Now, according to Pythagoras theorem,

$\bf \: AB^{2} + BC ^{2} = AC ^{2}$

In this case both the sides of square are equal and s = 8 cm,

so

Diagonal of square d :

$\bf \: d ^{2} = {8}^{2} + {8}^{2} \\ \bf \: {d}^{2} = 64 + 64 = 128 \\ \bf \: d = \sqrt{128} = 8\sqrt{2}$

So,

The diagonal of given square = 8√2 cm.

Answer 2

Answer:

A circle is inscribed in a square,

An equilateral triangle is inscribed in that circle.

Side of triangle = 4√3cm.

To find :

Length of diagonal of square.

Solution :

We know that side of an equilateral triangle = 4√3 cm.

This triangle is inscribed in a circle, it means this circle has a Circumradius.

The formula to find the Circumradius :

= \frac{ \textbf{side \: of \: triangle}}{ \sqrt{3} }=3side of triangle

So,

Circumradius r :

= \frac{4 \sqrt{3} }{ \sqrt{3} } = 4 \: cm=343=4cm

As we know that, circle is inscribed in a square,

so now we got the half length of side of that square in the form of Circumradius r.

so,

Side of square, s = 2r

s = 2 \times 4 = 8 \: cms=2×4=8cm

so,

side of square = 8 cm.

Now, according to Pythagoras theorem,

\bf \: AB^{2} + BC ^{2} = AC ^{2}AB2+BC2=AC2

In this case both the sides of square are equal and s = 8 cm,

so

Diagonal of square d :

\begin{gathered} \bf \: d ^{2} = {8}^{2} + {8}^{2} \\ \bf \: {d}^{2} = 64 + 64 = 128 \\ \bf \: d = \sqrt{128} = 8\sqrt{2} \end{gathered}d2=82+82d2=64+64=128d=128=82

So,

The diagonal of given square = 8√2 cm.