What is the radius of the incircle of the triangle whose sides measure 5, 12 and 13 units?
(A)2 units
(B)12 units
(C)6.5 units
(D)6 units
(E)7.5 units
Pretty straight forward problem. The triangle is a right triangle. For a right triangle, the radius of the incircle
r=Sum of perpendicular sides-hypotenuse/2 and u get the answer A.
Also,
What is the measure of the radius of the circle that circumscribes a triangle whose sides measure 9, 40 and 41?
(A)6
(B)4
(C)24.5
(D)20.5
(E)12.5
Another Pythagorean triplet with 41 as hypotenuse
In a right triangle, the radius of the circle that circumscribes the triangle is half the hypotenuse.
r=hypotenuse/2. Which gives the answer D
However, I remember these formulas from my school/college days and I don't recall seeing them in any of the gmat books I've been thru. It took me about 10-15 mins each to try and deduce/recall these formulas from memory. I've been thru the Manhattan gmat geometry guide 4th edition and there was little coverage on the circles topic. I guess my question is, are such problems tested frequently on the gmat. And should I be referring to a different theory book for comprehensive coverage on geometry eg Veritas Prep or Ez Solutions? Thanks
Geometry Problem
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- Brian@VeritasPrep
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Hey Knight,
You know, I wouldn't spend too much time worrying about the "obscure" rules like that incircle rule. The GMAT doesn't have much incentive to reward memorization like that - it's definitely not a test of how much you remember from high school geometry, but rather of your ability to leverage what you know about basic geometry (areas, perimeters, Pythagorean theorem, the coordinate plane) to solve seemingly unique problems.
So much about GMAT geometry is about finding ways to apply the basic things that you do know to these problems. So, for example, in your first question, without knowing or applying that incircle rule I could still draw the circle inside the triangle and apply a few basics.
Area = 1/2 bh = 1/2 5 * 12 = 30
So the "height" from the hypotenuse to the right angle is going to be:
A = 30 = 1/2 bh = 1/2 (13)(h)
So 30 = 1/2 (13h)
60 = 13h
h = something between 4 and 5
Well, the diameter of that circle is going to be less than that height, as the circle can't fill that entire space between the hypotenuse and the opposite corner (draw it and you'll see). And since the radius is "half of less than 4.5" by that logic, 2 is the only possible answer.
So while formulas like that one can be helpful, the GMAT doesn't require you to know anything that obscure...there's always a logical way to get the answer by seeing ways to use the basics. And in that way the basics area "sustainable" - trying to remember (or relearn) everything from high school geometry just isn't worth the time...especially because you know that that's not the GMAT's intent anyway.
That said, I would try to remember that "diameter + any point on the outside of a circle = right triangle". Why? Because it's a perfect link between circles and right triangles, and you know that the GMAT likes to make you blend shapes. But as far as pure formulas (and not just concepts) go, you're in great shape with perimeter/area/Pythagorean/coordinate plane, and some ingenuity to see where you can use those effectively.
You know, I wouldn't spend too much time worrying about the "obscure" rules like that incircle rule. The GMAT doesn't have much incentive to reward memorization like that - it's definitely not a test of how much you remember from high school geometry, but rather of your ability to leverage what you know about basic geometry (areas, perimeters, Pythagorean theorem, the coordinate plane) to solve seemingly unique problems.
So much about GMAT geometry is about finding ways to apply the basic things that you do know to these problems. So, for example, in your first question, without knowing or applying that incircle rule I could still draw the circle inside the triangle and apply a few basics.
Area = 1/2 bh = 1/2 5 * 12 = 30
So the "height" from the hypotenuse to the right angle is going to be:
A = 30 = 1/2 bh = 1/2 (13)(h)
So 30 = 1/2 (13h)
60 = 13h
h = something between 4 and 5
Well, the diameter of that circle is going to be less than that height, as the circle can't fill that entire space between the hypotenuse and the opposite corner (draw it and you'll see). And since the radius is "half of less than 4.5" by that logic, 2 is the only possible answer.
So while formulas like that one can be helpful, the GMAT doesn't require you to know anything that obscure...there's always a logical way to get the answer by seeing ways to use the basics. And in that way the basics area "sustainable" - trying to remember (or relearn) everything from high school geometry just isn't worth the time...especially because you know that that's not the GMAT's intent anyway.
That said, I would try to remember that "diameter + any point on the outside of a circle = right triangle". Why? Because it's a perfect link between circles and right triangles, and you know that the GMAT likes to make you blend shapes. But as far as pure formulas (and not just concepts) go, you're in great shape with perimeter/area/Pythagorean/coordinate plane, and some ingenuity to see where you can use those effectively.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
- prateek_guy2004
- Master | Next Rank: 500 Posts
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Brian@VeritasPrep great approach brian.....Its true and certainly logical way to solve that....
Knight- I did not know that formulae good to know that.....
Thanks mate
Knight- I did not know that formulae good to know that.....
Thanks mate
Don't look for the incorrect things that you have done rather look for remedies....
https://www.beatthegmat.com/motivation-t90253.html
https://www.beatthegmat.com/motivation-t90253.html