Equilateral Triangle Inscribed in a Circle

Q.

An equilateral triangle ABC is inscribed in a circle of radius r units, which is centered at O, as shown below. Then the length of the side of this triangle is equal to

1. $2r\sqrt{3}$

2. $r\sqrt{3}$

3. $\frac{r}{2}\sqrt{3}$

4. $\frac{r}{4}\sqrt{3}$

Q. ABC is a triangle inscribed in a circle, AC being the diameter of the circle. The length of AC is as much more than the length of BC as the length of BC is more than the length of AB. Find AC:AB

Q.

An equilateral triangle ABC of side 24 cm is inscribed in a circle which is centered at O, as shown below. Then the radius of this circle is

1. $8\sqrt{3}$ cm

2. $6\sqrt{3}$ cm

3. $4\sqrt{3}$ cm

4. $2\sqrt{3}$ cm

Q.

An equilateral triangle ABC of side 4 cm is inscribed in a circle which is centered at O, as shown below. Then the radius of this circle is

1. $\frac{5}{\sqrt{3}}$ cm

2. $\frac{4}{\sqrt{3}}$ cm

3. $\frac{2}{\sqrt{3}}$ cm

4. $\frac{1}{\sqrt{3}}$ cm

Q.

An equilateral triangle ABC is inscribed in a circle of radius 8 cm, which is centered at O, as shown below. Then the length of the side of this triangle is equal to

1. $8$ cm

2. $2\sqrt{3}$ cm

3. $4\sqrt{3}$ cm

4. $8\sqrt{3}$ cm

Q.

An equilateral triangle ABC is inscribed in a circle of radius 10 cm, which is centered at O, as shown below. Then the length of the side of this triangle is equal to

1. $25\sqrt{3}$ cm

2. $15\sqrt{3}$ cm

3. $10\sqrt{3}$ cm

4. $5\sqrt{3}$ cm

Q.

An equilateral triangle ABC of side 10 cm is inscribed in a circle which is centered at O, as shown below. Then the radius of this circle is

1. $\frac{10}{\sqrt{3}}$ cm

2. $\frac{8}{\sqrt{3}}$ cm

3. $\frac{4}{\sqrt{3}}$ cm

4. $\frac{5}{\sqrt{3}}$ cm

Q.

An equilateral triangle ABC is inscribed in a circle of radius 12 cm, which is centered at O, as shown below. Then the length of the side of this triangle is equal to

1. $4\sqrt{3}$ cm

2. $6\sqrt{3}$ cm

3. $12\sqrt{3}$ cm

4. $10\sqrt{3}$ cm

Q.

An equilateral triangle of side $9cm$ is inscribed in a circle. Find the radius of the circle.

Q.

An equilateral triangle ABC is inscribed in a circle of radius 18 cm, which is centered at O, as shown below. Then the length of the side of this triangle is equal to

1. $12\sqrt{3}$ cm

2. $18\sqrt{3}$ cm

3. $9\sqrt{3}$ cm

4. $3\sqrt{3}$ cm

Q.

A circle with Centre at $(2,4)$is such that the line $x+y+2=0$cuts a chord of length $6$.

The radius of the circle is

1. $\sqrt{41}$

2. $\sqrt{11}$

3. $\sqrt{21}$

4. $\sqrt{31}$

Q.

Two circles with centres, A and B and of radii 5 cm and 3 cm respectively touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle in P and Q. Find the length of PQ.

1. 5√6cm

2. 6√6cm

3. 4√6cm

4. 8√6cm

Q.

An equilateral triangle ABC is inscribed in a circle centered at O and of radius 9 cm, as shown below. OD is a radius such that it is perpendicular to BC and cuts BC at point E. Then the length of OE is

1. 5.5 cm

2. 5 cm

3. 6 cm

4. 4.5 cm

Q. The centroid of an equilateral triangle coincides with which of the following options?

1. Circumcentre
2. Orthocentre
3. Incentre
4. Altitude

Q.

A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distances from each other on its boundary, each having a toy telephone in his hand to talk to each other. Find the length of the string of each phone.

1. 40√3

2. 20√3

3. 30√3

4. 10√3

Q. An equilateral triangle $ABC,$ whose side is $6cm,$ is inscribed in a circle. Find the radius of the circle.

Q.

An equilateral triangle ABC of side 6 cm is inscribed in a circle which is centered at O, as shown below. Then the radius of this circle is

1. $2\sqrt{3}$ cm

2. $\frac{3}{4}\sqrt{3}$ cm

3. $\frac{1}{4}\sqrt{3}$ cm

4. $\sqrt{3}$ cm

Q. 14.let PQR be an equilateral triangle inscribed ina circle. if the distances of P, Q, R from S(a point on the circle) are l, m, n, find (l+m-n)(m+n-l)(n+l-m)+10

Q.

An equilateral triangle ABC is inscribed in a circle centered at O, as shown below. OE is a radius such that it is perpendicular to BC and cuts BC at point D. If OD = 2.2 cm, then the diameter of this circle is

1. 9.2 cm

2. 8.4 cm

3. 8.8 cm

4. 9.6 cm

Q. Which of the following statements are true and which are false? Give reasons for your answers.
$\left(i\right)$ Only one line can pass through a single point.
$\left(ii\right)$ There are an infinite number of lines which pass through two distinct points.
$\left(iii\right)$ A line can be produced indefinitely on both sides.
$\left(iv\right)$ If two circles are equal, then their radii are equal.
$\left(v\right)$ if $AB=PQ$ and $PQ=XY$, then $AB=XY$.

1. $\left(i\right),\left(ii\right)$ - True
   $\left(iii\right),\left(iv\right),\left(v\right)$ - False
2. $\left(i\right),\left(ii\right),\left(iii\right)$ - True
   $\left(iv\right),\left(v\right)$ - False
3. $\left(i\right),\left(ii\right)$ - False
   $\left(iii\right),\left(iv\right),\left(v\right)$ - True
4. $\left(i\right),\left(ii\right),\left(iii\right)$ - False
   $\left(iv\right),\left(v\right)$ - True
   

Q.

An equilateral triangle ABC is inscribed in a circle centered at O and of radius 4 cm, as shown below. OD is a radius such that it is perpendicular to BC and cuts BC at point E. Then the length of OE is

1. 0.5 cm

2. 2 cm

3. 3 cm

4. 1 cm

Q.

An equilateral triangle ABC is inscribed in a circle centered at O, as shown below. OE is a radius such that it is perpendicular to BC and cuts BC at point D. If OD = 7 cm, then the radius of this circle is

1. 22 cm

2. 18 cm

3. 14 cm

4. 10 cm

Q.

An equilateral triangle ABC of side $x$ units is inscribed in a circle which is centered at O, as shown below. Then the radius of this circle is

1. $\frac{x}{2\sqrt{3}}$ units

2. $\frac{x}{\sqrt{3}}$ units

3. $\frac{2x}{\sqrt{3}}$ units

4. $\frac{2x}{3\sqrt{3}}$ units

Q. $PQRS$ is a cyclic quadrilateral. $PR$ is the diameter of the circle. If $PQ=7cm,QR=6cm$ and $RS=cm$, then $PS=$:

1. $7cm$
2. $8cm$
3. $9cm$
4. $10cm$

Q.

An equilateral triangle ABC is inscribed in a circle centered at O and of radius 12 cm, as shown below. OD is a radius such that it is perpendicular to BC and cuts BC at point E. Then the length of OE is

1. 4 cm

2. 6 cm

3. 8 cm

4. 10 cm

Q.

An equilateral triangle $ABC,$ whose side is $6cm,$ is inscribed in a circle. Find the radius of the circle.

1. $2\sqrt{3}cm$

2. $2\sqrt{2}cm$

3. $3\sqrt{3}cm$

4. $3cm$

Q.

An equilateral triangle ABC is inscribed in a circle centered at O, as shown below. OE is a radius such that it is perpendicular to BC and cuts BC at point D. If OD = 5 cm, then the radius of this circle is

1. 15 cm

2. 12 cm

3. 7 cm

4. 10 cm

Q.

An equilateral triangle ABC is inscribed in a circle centered at O, as shown below. OE is a radius such that it is perpendicular to BC and cuts BC at point D. If OD = 3 cm, then the diameter of this circle is

1. 12 cm

2. 10 cm

3. 8 cm

4. 6 cm

Q. The cosine of the obtuse angle formed by medians drawn from the vertices of the acute angles of an isosceles right-angled triangle is $-\frac{k}{5}$, where $k=$

Q. If an isoceles triangle $ABC$ in which $AB=AC=6cm$ is inscribed in a circle of radius $9cm$, find the area of triangle.