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705-805 Level|   Geometry|                     
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kskumar
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A triangle has side lengths of a, b, and c centimeters. Does each angl 13 Oct 2021, 09:01
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avigutman Please add your view in this Problem
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A triangle has side lengths of a, b, and c centimeters. Does each angl 13 Oct 2021, 13:55
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avigutman Please add your view in this Problem

kskumar here's how I think through this problem.

Start with statement 2, as I don't know how it could possibly be enough to answer the question. You see, when I look at c+2, I ask myself what impact does the +2 have here? I don't even know whether it's 2cm or 2 feet or 2km... I have no way of knowing whether +2 is a lot or a little in the context of this triangle. The fact that a+b > c goes without saying, so the only new information provided by statement 2 is information for which I totally lack context and is therefore useless.
So, I've eliminated BD and I'm ready to evaluate statement 1.

Statement 1 provides the 2-dimensional ratio for the three sides (3:4:6), which I appreciate because that fits well with the Pythagorean theorem.
If the squares of the two smaller sides are 3 and 4 ratio units respectively, the Pythagorean theorem tells us that the square of the third side would have to be exactly 7 ratio units for this triangle to be a right triangle. Since it's less than 7 ratio units (it's 6 ratio units), I can conclude that the angle across from the longest side is less than 90 degrees, and the other angles have to be smaller still, so we get a definitive YES and I pick A.
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A triangle has side lengths of a, b, and c centimeters. Does each angl 08 Nov 2021, 02:02
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A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively.

(2) c < a + b < c + 2

I didnt know how to go about solving it, so I made a quick guess.
I reasoned that since stat 1 gives us areas, we can find the length of the diameter, and thus the lengths of all three sides. So now the triangle has fixed length sides. I assumed that if we make one angle obtuse, the length of the opposite side will change. Thus, we cant really change the shape of the triangle. It is sort of locked. Thus, there's a good chance that the angles will be locked in too. SO A is suff

For stat 2 I tested values, c=2 then a=2 and b=1 or c=3 then a=2 and b=2
Thus, the lengths of the sides cannot be locked into one particular length, SO, there's a good chance that B is insuff.

Now, does this all make sense, or I just got really lucky?

VeritasKarishma Bunuel MathRevolution egmat
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A triangle has side lengths of a, b, and c centimeters. Does each angl 09 Nov 2021, 01:38
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Chitra657
moonwalking
A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively.

(2) c < a + b < c + 2

I didnt know how to go about solving it, so I made a quick guess.
I reasoned that since stat 1 gives us areas, we can find the length of the diameter, and thus the lengths of all three sides. So now the triangle has fixed length sides. I assumed that if we make one angle obtuse, the length of the opposite side will change. Thus, we cant really change the shape of the triangle. It is sort of locked. Thus, there's a good chance that the angles will be locked in too. SO A is suff

For stat 2 I tested values, c=2 then a=2 and b=1 or c=3 then a=2 and b=2
Thus, the lengths of the sides cannot be locked into one particular length, SO, there's a good chance that B is insuff.

Now, does this all make sense, or I just got really lucky?

VeritasKarishma Bunuel MathRevolution egmat


Chitra657 - I am good with your statement 1 analysis. We know that we can find the exact length of all 3 sides. There is a unique triangle associated with 3 side lengths. The angles are all defined. Hence, knowing the side lengths, we know that the data is enough to answer the question. Whether our answer is "yes" or "no" is irrelevant. The point is that we can answer with a definite yes or no. Hence statement 1 is sufficient.

But for statement 2, though the sides are not uniquely defined, it is still possible that the given constraint leads to a unique answer.
Since acute angles have a^2 + b^2 > c^2, if the sides are constrained in a way that a^2 + b^2 > c^2, you will know that they represent an acute triangle. You don't need to know each side uniquely.
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A triangle has side lengths of a, b, and c centimeters. Does each angl 27 Oct 2022, 05:55
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moonwalking
A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively.

(2) c < a + b < c + 2

Statement 1: It effectively gives us measures of 3 sides. We can build a triangle with the sides and find out if any angle is greater than 90. So sufficient.
Statement 2: c < a + b stands for all triangles. a + b < c + 2 just says that sum of 2 sides is less than 3rd side +2. Not helpful as well. So not sufficient.

Therefore, Ans- A
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A triangle has side lengths of a, b, and c centimeters. Does each angl 10 Feb 2025, 09:59
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A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cm^2, 4 cm^2, and 6
[ltr]cm^2, respectively.

\(S(semicircle) = 1/2[color=#202122]π{r^2}\)

\(a^2 = 6/[color=#0f0f0f]π\)
\(b^2 = 8/π\)
\(c^2 = 12/π\)[/color][/color][/ltr]

[color=#202122]\(c^2 < a^2 + b^2\) —> acute triangle [/color]
Sufficient

(2) \(c < a + b < c + 2\)
\(c^2 < (a + b)^2 < (c + 2)^2\)
\(c^2 < a^2 + 2ab + b^2 < c^2 +4c + 4\)
Insufficient

Answer: A
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A triangle has side lengths of a, b, and c centimeters. Does each angl 10 Feb 2025, 21:34
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A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?

A triangle is acute if the square of the longest side is less than the sum of the squares of two smaller sides.

(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively.

Area of the semicircle with a cm as diameter \(= \pi a^2/8 = 3\)
Area of the semicircle with b cm as diameter \(= \pi b^2/8 = 4\)
Area of the semicircle with c cm as diameter \(= \pi c^2/8 = 6\)

Square of the longest side =\( c^2 = 48/\pi < a^2 + b^2 = 24/pi + 32/pi = 56\pi\)
The triangle is acute, i.e.each angle in the triangle measure less than 90 degrees
SUFFICIENT

(2) c < a + b < c + 2
a+b > c is always true
a+b<c+2
Case 1: a=2; b=3; c= 4; a^2 + b^2 = 2^2 + 3^2 = 13 < c^2 = 16; Triangle is NOT acute
Case 2: a=1; b=2; c=2; a^2 + b^2 = 1^2 + 2^2 = 5 > c^2 = 4; Triangle is acute.
NOT SUFFICIENT

IMO A
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