Question

10. The area of an equilateral triangle ABC is 17320.5
cm? With each vertex of the triangle as centre, a
circle is drawn with radius equal to half the length
of the side of the triangle (see Fig. 12.28). Find the
area of the shaded region. (Use it = 3.14 and
V3 = 1.73205)
Fig. 12.28​

Answer 1

Answer:

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Answer 2

$\LARGE{\bf{\underline{\underline{GIVEN:-}}}}$

• Area of equilateral triangle ΔABC = 17320.5 cm².
• A circle is drawn with radius equal to half the length  of the side of the triangle.

$\LARGE{\bf{\underline{\underline{TO \ FIND:-}}}}$

• Area of the shaded region

$\LARGE{\bf{\underline{\underline{SOLUTION:-}}}}$

Lets find the side of the triangle.

We know that area of equilateral triangle is given by:

$\boxed{\sf{\green{A_{equilateral}=\dfrac{\sqrt{3}}{4} \times a^2}}}$

Where:

• a is the side.

We know that area is 17320.5cm² and √3 = 1.73205. So substitute.

$\to \sf 17320.5=\dfrac{1.73205}{4} \times a^2$

$\to \sf \sqrt{17320.5 \times \dfrac{4}{1.73205}}= a$

$\to \sf \sqrt{ 4 \times 10000}= a$

$\to \sf{\green{200 \ cm= a}}$

Now, it is given that radius of the circles is 1/2 the length of side.

Therefore, radius can be given as:

$\boxed{\sf{\blue{Radius=\dfrac{1}{2} \times side. }}}$

$\sf \to R=\dfrac{1}{2} \times 200.$

$\to \sf{\green{ R=100 \ cm.}}$

Now find the area of sectors.

Since the radius is same for all circles, all the circles are congruent or identical. Therefore area of sector AYX, CXZ and BYZ are equal.

Area of sector:

$\boxed{\sf{\blue{\dfrac{\theta}{360} \times \pi \times r^2 }}}$

Where θ is angle made by sector, π is 3.14 and r is radius.

θ will be 60° because ∠A is 60° (Angles of an equilateral triangle are always 60°)

Substituting the values, we get:

$\to \sf \dfrac{60}{360} \times 3.14 \times 100^2$

$\to \sf{\green{\dfrac{15700}{3}cm^2}}$

Then area of three sectors = 3 × area of one sector.

$\to \sf \not{3} \times \dfrac{15700}{\not{3}}cm^2$

$\to \sf{\green{15700 cm^2}}$

Now, final step:

$\boxed{\sf{Area_{shaded \: region } = Area_{Equilateral \: triangle } - Area_{three \: sectors.}}}$

$\sf \to A=17320.5-15700$

$\leadsto \sf{\red{1620.5cm^2}} \bigstar$

Area of shaded region - 1620.5 cm²

Regards,

SujalSirimilla

Ex-brainly star.