## A triangle has side lengths of a, b, and c centimeters.

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### A triangle has side lengths of a, b, and c centimeters.

by Amzad Sadik » Tue Mar 20, 2018 9:27 pm
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

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by [email protected] » Wed Mar 21, 2018 5:09 pm
Hi Amzad Zadik,

For future reference, you should post your GMAT questions into the specific sub-forums. The Data Sufficiency forum can be found here:

https://www.beatthegmat.com/data-sufficiency-f7.html

We're told that a triangle has side lengths of A, B and C centimeters. We're asked if each angle in the triangle is LESS than 90 degrees. This is a YES/NO question. We can solve it with some Geometry facts and TESTing VALUES.

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.

Fact 1 tells us that the 3 side lengths of the triangle are 3, 4 and 6. IF the triangle had sides of 3, 4 and 5, then we would have a right triangle - and the largest angle would be 90 degrees. Since the '6 side' in this triangle is longer than the '5 side' in a 3/4/5 right triangle, the angle across from the '6 side' would be GREATER than 90 degrees. Thus, the answer to the question is ALWAYS NO.
Fact 1 is SUFFICIENT.

2) C < A+B < C+2

With the information in Fact 2, we can TEST VALUES.

IF.... A=3, B=4, C=6, then we know that the answer is NO (we proved it in Fact 1).
IF... A=1.9, B=1.9, C=1.9, then we have an EQUILATERAL triangle and all three angles are 60 degrees - and the answer is YES.
Fact 2 is INSUFFICIENT.

Final Answer: A

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by [email protected] » Mon Jan 21, 2019 10:41 am
Amzad Sadik wrote: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
Target question: Does each angle in the triangle measure less than 90 degrees?

Given: A triangle has side lengths of a, b, and c centimeters

Statement 1: The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cmÂ², 4 cmÂ², and 6 cmÂ², respectively.
This statement illustrates an important rule concerning Data Sufficiency questions: DO NOT DO MORE WORK THAN IS NECESSARY
Since we have the area for each circle, we COULD calculate the diameter of each circle.
So, after some tedious calculations, we WOULD know the lengths of all 3 sides of the triangle.
IF we know the lengths of all 3 sides of the triangle, we COULD draw the triangle, and we COULD then measure each angle with a protractor.
This means we COULD definitely determine whether each angle is less than 90 degrees.
In other words, we COULD answer the target question with certainty.
This means statement 1 is SUFFICIENT

Statement 2: c < a + b < c + 2
There are several values of a, b and c that satisfy statement 2. Here are two:
Case a: a = 1, b = 1 and c = 1. In this case, we have an EQUILATERAL triangle in which all three angles measure 60 degrees. So, the answer to the target question is YES, each angle in the triangle measures less than 90 degrees
Case b: a = 1.5, b = 2 and c = 2.5.
ASIDE: We know that a 3-4-5 triangle is a RIGHT triangle.
If we make each side HALF as long, we get a 1.5-2-2.5 triangle, which is also a RIGHT triangle.
In this case, we have a RIGHT triangle which means one angle EQUALS 90 degrees. So, the answer to the target question is NO, it is NOT the case that each angle in the triangle measures less than 90 degrees
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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by [email protected] » Mon Jan 21, 2019 4:06 pm
Amzad Sadik wrote: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
$$?\,\,\,\,\,\,:\,\,\,\,\,\,{a^2} < {b^2} + {c^2}\,\,\,\,\,AND\,\,\,\,\,{b^2} < {a^2} + {c^2}\,\,\,\,\,AND\,\,\,\,\,{c^2} < {a^2} + {b^2}\,\,\,\,\,\,\,\,\,\,\,\left[ {\,\Delta = \left( {a,b,c} \right)\,\,\,{\rm{side}}\,\,{\rm{lengths}}\,} \right]$$

$$\left( 1 \right)\,\,{\rm{lengths}}\,\,{\rm{of}}\,\,{\rm{sides}}\,\,{\rm{are}}\,\,\left( {{\rm{uniquely}}} \right)\,\,{\rm{known}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}$$

(We call this a "blind decision", because we know the answer is unique although we wonÂ´t need to discover if it is a "Yes" or if it is a "No"!)

$$\left( 2 \right)\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a;b;c} \right) = \left( {1;2;2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr \,{\rm{Take}}\,\,\left( {a;b;c} \right) = \left( {3 \cdot {1 \over 2};4 \cdot {1 \over 2};5 \cdot {1 \over 2}} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left[ {{c^2} = {a^2} + {b^2}} \right] \hfill \cr} \right.$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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### Re: A triangle has side lengths of a, b, and c centimeters.

by [email protected] » Fri Jun 11, 2021 3:43 pm
Amzad Sadik wrote:
Tue Mar 20, 2018 9:27 pm
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
Solution:

We need to determine whether each angle of a triangle whose side lengths are a, b and c centimeters measures less than 90 degrees. That is, we need to determine whether the triangle is an acute triangle. Notice that if c is the longest side of the triangle, then the triangle is an acute triangle if and only if a^2 + b^2 > c^2. Therefore, we need to determine whether a^2 + b^2 > c^2.

Statement One Alone:

If we let a < b < c, from statement one, we see that π(a/2)^2 / 2 = 3, π(b/2)^2 / 2 = 4 and π(c/2)^2 / 2 = 6. Let’s simplify the first equation:

π(a/2)^2 = 6

a^2 / 4 = 6 / π

a^2 = 24 / π

Similarly, if we simplify the other two equations, we will have b^2 = 32/π and c^2 = 48/π. Since a^2 + b^2 = 24/π + 32/π = 60/π is greater than c^2 = 48/π, we see that the triangle is indeed an acute triangle. Statement one alone is sufficient.

Statement Two Alone:

Statement two alone is not sufficient. If a = 1 and b = 1.8 and c = 2, then a^2 + b^2 = 1^2 + 1.8^2 = 1 + 3.24 = 4.24 is greater than c^2 = 4. In this case, the triangle is an acute triangle. However, if a = 1, b = 1.7 and c = 2, then a^2 + b^2 = 1^2 + 1.7^2 = 1 + 2.89 = 3.89 is not greater than c^2 = 4. In this case, the triangle is not an acute triangle.

Answer: A

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