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9, 40 and 41

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by sanju09 » Tue Feb 24, 2009 4:39 am
What is the radius of the circle that circumscribes the triangle, measure of whose sides is 9, 40 and 41?

A. 20

B. 20.5

C. 4

D. 8

E. 45
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by sureshbala » Tue Feb 24, 2009 4:52 am
The given triangle is a right angled triangle whose hypotenuse is 41. In a right angled triangle, Circumradius = 1/2 (hypotenuse) = 20.5
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by mjjking » Tue Feb 24, 2009 6:32 am
can you elaborate a bit please?
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by DanaJ » Tue Feb 24, 2009 6:47 am
You have lengths of sides 9, 40 and 41. Consider the squares:
9^2 = 81
40^2 = 1600
41^2 = 1681

As you can see, adding 9^2 to 40^2 gives you 41^2, so this is a right triangle, with a 41 hypotenuse. Now, right triangles have a series of properties and we will use the fact that the radius that circumscribes such a triangle is equal to half the hypotenuse (this is a commonly used property- you might want to remember it). This means that the length you are looking for is 41/2 = 20.5.
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by ven4gmat » Tue Feb 24, 2009 7:00 am
Now I could recollect this point which I have learnt in my school days...

In a right angled triangle, the right angled vertex is the Orthocentre and the midpoint of the hypotenuse is the Circumcentre.

Thanq Sureshbala and DanaJ
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by marcusking » Tue Feb 24, 2009 9:36 am
ven4gmat wrote:Now I could recollect this point which I have learnt in my school days...

In a right angled triangle, the right angled vertex is the Orthocentre and the midpoint of the hypotenuse is the Circumcentre.

Thanq Sureshbala and DanaJ
I think you mean the midpoint of the hypotenuse is the radius.
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by DanaJ » Tue Feb 24, 2009 9:49 am
A midpoint is only a point (in this case the point equally distant from both ends of the hypotenuse), it cannot be a radius. The radius we are looking for is half the length of the hypotenuse.
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by roger_michael » Tue Feb 24, 2009 9:51 am
Do you guys know where i can find all these "triangle inside a circle" rules? Any geometry book will have them? Thanks!
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by dendude » Tue Feb 24, 2009 10:11 am
marcusking wrote:I think you mean the midpoint of the hypotenuse is the radius.
DanaJ wrote:A midpoint is only a point (in this case the point equally distant from both ends of the hypotenuse), it cannot be a radius. The radius we are looking for is half the length of the hypotenuse.
I think this is derived from the rule that a right-triangle within a circle has the diameter as one of its sides or vice versa (or something like this that I can't recollect :?)
If the diameter is the hypotenuse, then half of it will be the radius.
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by sanju09 » Wed Feb 25, 2009 2:05 am
roger_michael wrote:Do you guys know where i can find all these "triangle inside a circle" rules? Any geometry book will have them? Thanks!
Collect all your school math books from standard VI to X and see for the basics in those books. This will help you in almost everything on GMAT CAT Quant :)
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