Question

Hey brilliance!
1) the number of tangent that can be drawn to a circle at point of the circle is_______.
2) two circles of radii 5.5 cm and 3.3 cm respectively touch each other.
what is the distance between the centre?
3)out of the following which is the Pythagorean triplet?




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Answer 1

Solution :- 1

Tangent is a line which touches the circle exactly at one point.

So,

The number of tangent that can be drawn to a circle at point of the circle is 1.

Hence,

$\: \: \: \: \: \: \: \: \red{ \boxed{ \bf \: Option \: (B) \: is \: correct}}$

Solution:- 2

Basic Concept :-

Two circles of radius r and R (where R > r) touch each other either internally or externally

• If two circles touch externally, then distance between their centres is R + r units

And

• If two circles touch internally, then distance between their centres is R - r units

Now,

We have provided two circles of radius 5.5 cm and 3.3 cm

So,

• If two circle touches externally, then distance between their centres = 5.5 + 3.3 = 8.8 cm

and

• If two circles touch internally, then distance between their centres = 5.5 - 3.3 = 2.2 cm

Hence,

$\: \: \: \: \: \: \: \: \red{ \boxed{ \bf \: Option \: (A) \: is \: correct}}$

Solution :- 3

To check the pythagorean triplet, we use the concept of Converse of Pythagoras Theorem

If the square of longest side is equal to sum of the squares of remaining two sides, then it forms a right angle triangle and hence formed pythagorean triplet.

(a) 5, 12, 14

$\rm :\longmapsto\: {14}^{2} = {5}^{2} + {12}^{2}$

$\rm :\longmapsto\: 196 = 25 + 144$

$\rm :\longmapsto\: 196 = 169$

$\rm :\longmapsto\:which \: is \: not \: possible$

(b) 3, 4, 2

$\rm :\longmapsto\: {4}^{2} = {3}^{2} + {2}^{2}$

$\rm :\longmapsto\:16 = 9 + 4$

$\rm :\longmapsto\:16 = 13$

$\rm :\longmapsto\:which \: is \: not \: possible$

(c) 8, 5, 17

$\rm :\longmapsto\: {17}^{2} = {15}^{2} + {8}^{2}$

$\rm :\longmapsto\:289 = 225 + 64$

$\rm :\longmapsto\:289 = 289$

$\pink{\bf :\longmapsto\:Hence, \: (8, 5, 17) \: is \: pythagorean \: triplet}$

$\: \: \: \: \: \: \: \: \red{ \boxed{ \bf \: Option \: (C) \: is \: correct}}$

Solution :- 4

Concept Used :-

Area Ratio Theorem :-

• This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

Given that

$\red{\rm :\longmapsto\: \bf \: \triangle ABC \: \sim \:\triangle PQR }$

$\red{\rm :\longmapsto\: \bf \: ar(\triangle \: ABC) \: = 25}$

$\red{\rm :\longmapsto\: \bf \: ar(\triangle \: PQR) \: = 16}$

So,

By Area Ratio Theorem,

$\rm :\longmapsto\:\dfrac{ar(\triangle ABC)}{ar(\triangle PQR)} = \dfrac{ {AB}^{2} }{ {PQ}^{2} }$

$\rm :\longmapsto\:\dfrac{25}{16} = \dfrac{ {AB}^{2} }{ {PQ}^{2} }$

$\bf\implies \:\dfrac{AB}{PQ} = \dfrac{5}{4}$

$\purple{ \: \bf\implies \:AB : PQ \: = \: 5 : 4}$

Hence,

$\: \: \: \: \: \: \: \: \red{ \boxed{ \bf \: Option \: (D) \: is \: correct}}$

Additional Information :-

1. Pythagoras Theorem :-

• This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

• This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

• This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4 Basic Proportionality Theorem,

• If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

Answer 2

Refer the attachment..!!

Is this ??