Circles: Three circles fit inside rectangle as shown....

S

sojeee

New member
Joined
Oct 24, 2017
Messages
25
m.jpg Three circles fit inside rectangle as shown.
Two of the circles are identical and the third is larger.
The circles have radii 9cm, 9cm and 25cm.

Calculate the lengh, l, of the rectangle.
 
sojeee said:
View attachment 9343 Three circles fit inside rectangle as shown.
Two of the circles are identical and the third is larger.
The circles have radii 9cm, 9cm and 25cm.

Calculate the lengh, l, of the rectangle.
Click to expand...

I'm not going to suggest a solution, since you have shown no work, but I will suggest that you draw 11 different radii. I won't tell you which ones. Maybe it will lead you somewhere. Let's see what you get.
 
Last edited:
tkhunny said:
I'm not going o suggest a solution, since you have shown no work, but I will suggest that you draw 11 different radii. I won't tell you which ones. Maybe it will lead you somewhere. Let's see what you get.
Click to expand...

could you please explain what you mean by 11 different radii
 
sojeee said:
could you please explain what you mean by 11 different radii
Click to expand...

A radius, in the sense intended by tkhunny, is a segment from the center of a circle to its circumference. In your picture, there are 11 places where you can do this, from the center of a circle to a point shown on the picture. For example, there are 5 points on the large circle where it touches something.

If you draw these 11 radii, labeling them with their lengths, you should be able to write an equation or two with which you can find lengths of parts of the rectangle. The Pythagorean theorem will be particularly useful.

Give that a try, and let us know your results. Keep in mind that we tend to be much more helpful when you show work, which helps us know what help you need.
 
Dr.Peterson said:
A radius, in the sense intended by tkhunny, is a segment from the center of a circle to its circumference. In your picture, there are 11 places where you can do this, from the center of a circle to a point shown on the picture. For example, there are 5 points on the large circle where it touches something.

If you draw these 11 radii, labeling them with their lengths, you should be able to write an equation or two with which you can find lengths of parts of the rectangle. The Pythagorean theorem will be particularly useful.

Give that a try, and let us know your results. Keep in mind that we tend to be much more helpful when you show work, which helps us know what help you need.
Click to expand...

l.jpg
The yellow part is what I cannot figure out
 
Yes, the 2 middle segments are not as clear as the other 2. The Pythagorean theorem mentioned earlier would be helpful here. Draw some additional lines until you get a shape to which the theorem can be applied.
 
sojeee said:
View attachment 9344
The yellow part is what I cannot figure out
Click to expand...

I wouldn't draw the yellow line at all. The length you need to find can be seen as the altitude of a right-angled trapezoid with sides 25, 25+9, and 9. If you draw another line in there, you will be able to use Pythagoras.
 
Dr.Peterson said:
I wouldn't draw the yellow line at all. The length you need to find can be seen as the altitude of a right-angled trapezoid with sides 25, 25+9, and 9. If you draw another line in there, you will be able to use Pythagoras.
Click to expand...

Could you please direct me as to where to draw this extra line in the trapezoid
 
With a little experience, you can just draw the radii you need, rather than all 11.
 
sojeee said:
Could you please direct me as to where to draw this extra line in the trapezoid
Click to expand...

The mention of the Pythagorean theorem is a major hint. How can you make a right triangle, two of whose sides you know?

Additional hint: the length of this line will be the missing part of the length L.
 
Dr.Peterson said:
The mention of the Pythagorean theorem is a major hint. How can you make a right triangle, two of whose sides you know?

Additional hint: the length of this line will be the missing part of the length L.
Click to expand...

Thankyou for the help I have worked it out and got 64cm in total
 
arkvee said:
Thankyou for the help I have worked it out and got 64cm in total
Click to expand...

That's right, so I assume you found the right triangle.

This trick is used in many geometry problems involving either circles or trapezoids; for example, it is used to find the length of a common tangent between two circles. Part of the reason for giving only hints is to help this way of thinking become more memorable for you!
 
You must log in or register to reply here.
Top