Geometrical Method to Find Square Root of a Number

Q.

PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm. If PQ = 16 cm and RS = 12 cm, find the distance between PQ and RS, if they lie.
(i) on the same side of the centre O.

(ii) on opposite of the centre O.

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Q. Simplify : $5\sqrt[3]{3250}+7\sqrt[3]{316}-14\sqrt[3]{354}$

Q.

If $x=3-2\sqrt{2}$ then $(x+\frac{1}{x}{)}^3$ = __________________

1. 343

2. 216

3. 125

4. 27

Q.

If $x=3-2\sqrt{2}$ , then $\frac{1}{x}=$ _____.

1. $3+2\sqrt{2}$

2. $3-2\sqrt{2}$

3. $6\sqrt{2}$

4. $6+2\sqrt{2}$

Q. Question 6
Represent geometrically the following number on the number line
(i) $\sqrt{4.5}$
(ii) $\sqrt{5.6}$
(iii) $\sqrt{8.1}$
(iv) $\sqrt{2.3}$
Thinking process
(i) Firstly, we draw a line segment AB of length equal to the number inside the root and extend it to C such that BC = 1.
(ii) Draw a semi-circle with centre O ( O is the midpoint of AC ) and radius OA.
(iii) Now, draw a perpendicular line from B to cut the semicircle at D.

(iv)Further, draw an arc with centre B and radius BD meeting AC produced at E.

Q.

Find the square root of $\sqrt{24}$ using geometric method.

1. 3.9

2. 2.4

3. 4.9

4. 6.4

Q. Find the square root of $\sqrt{28}$ using geometric method.

1. 6.4
2. 4.9
3. 3.9
4. 5.3

Q.

Construct a traingle of sides 5 cm, 8 cm, 12 cm and then construct a parallelogram whose area is equal to area of that triangle and one of whose angles is ${60}^{\circ }$. The dimensions of such parallelogram are: ___ (approximately).

1. 9 cm, 4 cm

   9 cm, 4 cm

2. 6 cm, 2.6 cm

   6 cm, 2.6 cm

3. 9 cm, 12 cm

   9 cm, 12 cm

4. 4 cm, 3 cm

   4 cm, 3 cm

Q.

Find the value of $3\sqrt{48}-4\sqrt{75}+5\sqrt{192}$

1. $32\sqrt{3}$

2. $7\sqrt{3}$

3. $24\sqrt{3}$

4. $42\sqrt{3}$

Q.

The numerical expression $\frac{3}{8}+\frac{-5}{7}$ equal to $\frac{-19}{56}$ states that

1. rational numbers are not closed under subtraction

2. addition of rational numbers is not commutative

3. rational numbers are closed under addition

4. addition of rational numbers is commutative

Q.

In the given figure, diameter of the semi-circle is $6cm$ and $CD=1cm$.

Find $AB$

1. None of these

2. $\sqrt{15}$

3. $\sqrt{30}$

4. $\sqrt{31}$

Q. Find the value of $\frac{3}{5}+\frac{-7}{6}+\frac{2}{5}+\frac{-5}{6}$.

Q.

Find the square root of 10 using geometric method.

1. 4

2. 1

3. 2.5

4. 3.2

Q.

Rationalize the denominator of the following:

$\frac{1}{(\sqrt{7}-\sqrt{6})}$

Q. In the figure, $O$ is the centre of the circle of radius $5\text{cm}.$ $P$ and $Q$ are points on chords $AB$ and $CD$ respectively such that $OP\perp AB,OQ\perp CD,AB||CD,AB=6\text{cm}$ and $CD=8cm.$ Determine $PQ$.

Q. A circle has radius $\sqrt{2}$ cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in the major segment is ${45}^{\circ }$.

Q.

$(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})=1$

1. Product of two irrational numbers is a rational number

2. Irrational number has a non- terminating recurring decimal expansion

Q.

Find the value of $\sqrt[5]{516}\times \sqrt[5]{52}$.

1. 32

2. $\sqrt[5]{518}$

3. 4

4. 2

Q. AB and CD are equal chords. If OE is 3 cm and AB is 8 cm, then the value of OC is .

1. 4 cm
2. 6 cm
3. 5 cm
4. 3 cm

Q. A circle has radius $\sqrt{2}$ cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in the major segment is ${45}^{\circ }$.

Q. In the figure, $O$ is the centre of the circle of radius $5\text{cm}.$ $P$ and $Q$ are points on chords $AB$ and $CD$ respectively such that $OP\perp AB,OQ\perp CD,AB||CD,AB=6\text{cm}$ and $CD=8cm.$ Determine $PQ$.

Q.

The numerical expression $\frac{3}{8}$ + ($\frac{-5}{7}$) equal to $\frac{-19}{56}$ states that:

1. rational numbers are not closed under subtraction

2. addition of rational numbers is not commutative

3. rational numbers are closed under addition

4. addition of rational numbers is commutative

Q. Two parallel chords $AB$ and $CD$ are drawn on the same side of centre $O$ of a circle. Radius of circle is $65m$, length of chords are $112m$ and $126m$. Find the area of $\square ABCD$.

1. $2023m^2$
2. $2123m^2$
3. $2223m^2$
4. $2323m^2$

Q. A circle has radius $\sqrt{2}$ cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in the major segment is ${45}^{\circ }$.

Q.

If $x=3-2\sqrt{2}$ , then $\frac{1}{x}=$ _____.

1. $3+2\sqrt{2}$

2. $3-2\sqrt{2}$

3. $6\sqrt{2}$

4. $6+2\sqrt{2}$

Q.

The expression $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ when simplified reduces to a-b$\sqrt{15}$.

Then a + b is __________.

1. 15

2. 10

3. 20

4. 5

Q.

If $x=3-2\sqrt{2}$ , then $\frac{1}{x}=$ _____.

1. $3+2\sqrt{2}$

2. $3-2\sqrt{2}$

3. $6\sqrt{2}$

4. $6+2\sqrt{2}$

Q.

Find the value of $\sqrt{4}$ using geometric method.

1. 2

2. 1

3. 2.1

4. 2.9

Q.

$\sqrt{\frac{3}{49}}$ is an irrational number.

1. True

2. False

Q.

If $x=3-2\sqrt{2}$ then $(x+\frac{1}{x}{)}^3$ = __________________