edited by
578 views
1 votes
1 votes

$\text{AB}$ is a diameter of a circle of radius $5$ cm. Let $\text{P}$ and $\text{Q}$ be two points on the circle so that the length of $\text{PB}$ is $6$ cm, and the length of $\text{AP}$ is twice that of $\text{AQ}$. Then the length, in cm, of $\text{QB}$ is nearest to

  1. $7.8$
  2. $8.5$
  3. $9.1$
  4. $9.3$
edited by

1 Answer

1 votes
1 votes

We can draw, the below diagram:

Let $`x\text{’}$ be the length of $ \text{AQ},$ and  $`y\text{’}$ be the length of $\text{QB}.$

Given that,

Radius of a circle $ = 5 \; \text{cm} $

And, $ \text{AB}$ is a diameter of a circle $ = 2 \times \text{radius of a circle} = 10 \; \text{cm} $

Let $ \text{P}$ and $ \text{Q}$ be two points on the circle.

So, the length of $ \text{PB} = 6 \; \text{cm} $

The length of $\text{AP}$ is twice that of length $\text{AQ}$.

So, the length of $ \text{AP} = 2x $

We can take out the $\triangle \text{APB},$ from the above diagram:



Apply the Pythagoras’ theorem $ : \boxed{\text{Hypotenuse}^{2} = \text{Perpendicular}^{2} + \text{Base}^{2}} $

$ (10)^{2} = (2x)^{2} + (6)^{2} $

$ \Rightarrow 100 = 4x^{2} + 36 $

$ \Rightarrow 100 – 36 = 4x^{2} $

$ \Rightarrow 64 = 4x^{2} $

$ \Rightarrow 16 = x^{2} $

$ \Rightarrow \sqrt{16} = x $

$ \Rightarrow \boxed{x = 4}$

Now, We can take out the $\triangle \text{AQB},$ from the first diagram:
 



Again apply the Pythagoras’ theorem :

$ (10)^{2} = (4)^{2} + (y)^{2} $

$ \Rightarrow 100 = 16 + y^{2} $

$ \Rightarrow 84 = y^{2} $

$ \Rightarrow \sqrt{84} = y $

$ \Rightarrow \boxed{y = 9.16} $

Correct Answer $: \text{C} $


$\textbf{PS:}$ Thales's theorem: if $AC$ is a diameter and $B$ is a point on the diameter's circle, the angle $ABC$ is a right angle.

References: 

edited by
Answer:

Related questions

1 votes
1 votes
1 answer
1
go_editor asked Mar 8, 2020
797 views
Corners are cut off from an equilateral triangle T to produce a regular hexagon H. Then, the ratio of the area of H to the area of T is $5:6$$4:5$$3:4$$2:3$
1 votes
1 votes
1 answer
2
go_editor asked Mar 8, 2020
592 views
In a circle of radius $11$ cm, CD is a diameter and AB is a chord of length $20.5$ cm. If AB and CD intersect at a point E inside the circle and CE has length $7$ cm, the...
2 votes
2 votes
1 answer
3
go_editor asked Mar 8, 2020
592 views
With rectangular axes of coordinates, the number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$, where each step from any point $(x,y)$ is either to $(x,y+1)$ or to $(x+1,...
2 votes
2 votes
1 answer
4
go_editor asked Mar 8, 2020
516 views
Let T be the triangle formed by the straight line $3x+5y-45=0$ and T the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as ...
1 votes
1 votes
1 answer
5