Calculating the length of the side of an acute / obtuse triangle - GMAT Quantitative
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The sides of a triangle are 4, 8, and an integer 
. How many possible values does 
have?
The sides of a triangle are 4, 8, and an integer
If two sides are 4 and 8, then the third side must be greater than 
and less than 
. This means 
can be 5, 6, 7, 8, 9, 10, or 11.
If two sides are 4 and 8, then the third side must be greater than
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Two sides of a triangle measure 5 inches and 11 inches. Which of the following statements correctly expresses the range of possible lengths of the third side 
?
Two sides of a triangle measure 5 inches and 11 inches. Which of the following statements correctly expresses the range of possible lengths of the third side
By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.
Case 1: 
is the greatest of the three sidelengths.
Then 
Case 2: 
is not the greatest of the three sidelengths - that is, 11 is.
Then 
, or, equivalently, 
.
Therefore, 
.
By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.
Case 1:
Then
Case 2:
Then
Therefore,
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Which of the following is true of a triangle with sides that measure 15, 17, and 21?
Which of the following is true of a triangle with sides that measure 15, 17, and 21?
The triangle can exist by the Triangle Inequality, since the sum of the two smaller sides exceeds the greatest:
To determine whether the triangle is acute, right, or obtuse, add the squares of the two smaller sides, and compare the sum to the square of the largest side.
Since this sum is greater, the triangle is acute.
The triangle can exist by the Triangle Inequality, since the sum of the two smaller sides exceeds the greatest:
To determine whether the triangle is acute, right, or obtuse, add the squares of the two smaller sides, and compare the sum to the square of the largest side.
Since this sum is greater, the triangle is acute.
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Let the three interior angles of a triangle measure 
, and 
. Which of the following statements is true about the triangle?
Let the three interior angles of a triangle measure
If these are the measures of the interior angles of a triangle, then they total 
. Add the expressions, and solve for 
.
One angle measures 
The others measure:
Since the largest angle measures greater than 
, the angle is obtuse, and the triangle is as well. Since the three angles each have different measure, their opposite sides do also, making the triangle scalene.
If these are the measures of the interior angles of a triangle, then they total
One angle measures
Since the largest angle measures greater than
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In 
, 
and 
. Which of the following values of 
makes 
a scalene triangle?
In
The three sides of a scalene triangle have different measures, so 15 can be eliminated.
By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since 
, 8 violates this theorem; since 
, 22 does as well.
10 is a valid measure of the third side, since 
; it makes all three segments of different length, so it is the correct choice.
The three sides of a scalene triangle have different measures, so 15 can be eliminated.
By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since
10 is a valid measure of the third side, since
Compare your answer with the correct one above
is a scalene triangle with perimeter 30. 
. Which of the following cannot be equal to 
?
The three sides of a scalene triangle have different measures. One measure 
cannot have is 12, but this is not a choice.
It cannot be true that 
. Since the perimeter is
, we can find out what other value can be eliminated as follows:
Therefore, if , then 
, and the triangle is not scalene. 9 is the correct choice.
The three sides of a scalene triangle have different measures. One measure
It cannot be true that
Therefore, if
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is a scalene triangle with perimeter 33; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?
By trial and error, we get four ways to add distinct primes to yield sum 33:
In each case, however the Triangle Inequality is violated - the sum of the two shortest lengths does not exceed the third.
No triangle can exist as described.
By trial and error, we get four ways to add distinct primes to yield sum 33:
In each case, however the Triangle Inequality is violated - the sum of the two shortest lengths does not exceed the third.
No triangle can exist as described.
Compare your answer with the correct one above
is an isosceles triangle with perimeter 43; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?
We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).
We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. 
, 
, and 
include numbers that are not prime (21, 15, 9). This leaves us with only one possibility:
- greatest length 19
19 is the correct choice.
We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).
We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three.
19 is the correct choice.
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The lengths of the sides of a scalene triangle are all prime numbers, and so is the perimeter of the triangle. What is the least possible perimeter of the triangle?
The lengths of the sides of a scalene triangle are all prime numbers, and so is the perimeter of the triangle. What is the least possible perimeter of the triangle?
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer.
One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:
- incorrect
- correct
The correct answer, 19, presents itself quickly.
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer.
One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:
The correct answer, 19, presents itself quickly.
Compare your answer with the correct one above
is a scalene triangle with perimeter 47; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47.
There are ten ways to add three distinct primes to yield sum 47:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:
The greatest possible length of the longest side is 23.
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47.
There are ten ways to add three distinct primes to yield sum 47:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:
The greatest possible length of the longest side is 23.
Compare your answer with the correct one above
The sides of a triangle are 4, 8, and an integer . How many possible values does have?
The sides of a triangle are 4, 8, and an integer
If two sides are 4 and 8, then the third side must be greater than and less than . This means can be 5, 6, 7, 8, 9, 10, or 11.
If two sides are 4 and 8, then the third side must be greater than
Compare your answer with the correct one above
Two sides of a triangle measure 5 inches and 11 inches. Which of the following statements correctly expresses the range of possible lengths of the third side ?
Two sides of a triangle measure 5 inches and 11 inches. Which of the following statements correctly expresses the range of possible lengths of the third side
By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.
Case 1: is the greatest of the three sidelengths.
Then
Case 2: is not the greatest of the three sidelengths - that is, 11 is.
Then , or, equivalently, .
Therefore, .
By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.
Case 1:
Then
Case 2:
Then
Therefore,
Compare your answer with the correct one above
Which of the following is true of a triangle with sides that measure 15, 17, and 21?
Which of the following is true of a triangle with sides that measure 15, 17, and 21?
The triangle can exist by the Triangle Inequality, since the sum of the two smaller sides exceeds the greatest:
To determine whether the triangle is acute, right, or obtuse, add the squares of the two smaller sides, and compare the sum to the square of the largest side.
Since this sum is greater, the triangle is acute.
The triangle can exist by the Triangle Inequality, since the sum of the two smaller sides exceeds the greatest:
To determine whether the triangle is acute, right, or obtuse, add the squares of the two smaller sides, and compare the sum to the square of the largest side.
Since this sum is greater, the triangle is acute.
Compare your answer with the correct one above
Let the three interior angles of a triangle measure , and . Which of the following statements is true about the triangle?
Let the three interior angles of a triangle measure
If these are the measures of the interior angles of a triangle, then they total . Add the expressions, and solve for .
One angle measures The others measure:
Since the largest angle measures greater than , the angle is obtuse, and the triangle is as well. Since the three angles each have different measure, their opposite sides do also, making the triangle scalene.
If these are the measures of the interior angles of a triangle, then they total
One angle measures
Since the largest angle measures greater than
Compare your answer with the correct one above
In , and . Which of the following values of makes a scalene triangle?
In
The three sides of a scalene triangle have different measures, so 15 can be eliminated.
By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since , 8 violates this theorem; since , 22 does as well.
10 is a valid measure of the third side, since ; it makes all three segments of different length, so it is the correct choice.
The three sides of a scalene triangle have different measures, so 15 can be eliminated.
By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since
10 is a valid measure of the third side, since
Compare your answer with the correct one above
is a scalene triangle with perimeter 30. . Which of the following cannot be equal to ?
The three sides of a scalene triangle have different measures. One measure cannot have is 12, but this is not a choice.
It cannot be true that . Since the perimeter is
, we can find out what other value can be eliminated as follows:
Therefore, if , then , and the triangle is not scalene. 9 is the correct choice.
The three sides of a scalene triangle have different measures. One measure
It cannot be true that
Therefore, if
Compare your answer with the correct one above
is a scalene triangle with perimeter 33; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?
By trial and error, we get four ways to add distinct primes to yield sum 33:
In each case, however the Triangle Inequality is violated - the sum of the two shortest lengths does not exceed the third.
No triangle can exist as described.
By trial and error, we get four ways to add distinct primes to yield sum 33:
In each case, however the Triangle Inequality is violated - the sum of the two shortest lengths does not exceed the third.
No triangle can exist as described.
Compare your answer with the correct one above
is an isosceles triangle with perimeter 43; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?
We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).
We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. , , and include numbers that are not prime (21, 15, 9). This leaves us with only one possibility:
- greatest length 19
19 is the correct choice.
We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).
We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three.
19 is the correct choice.
Compare your answer with the correct one above
The lengths of the sides of a scalene triangle are all prime numbers, and so is the perimeter of the triangle. What is the least possible perimeter of the triangle?
The lengths of the sides of a scalene triangle are all prime numbers, and so is the perimeter of the triangle. What is the least possible perimeter of the triangle?
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer.
One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:
- incorrect
- correct
The correct answer, 19, presents itself quickly.
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer.
One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:
The correct answer, 19, presents itself quickly.
Compare your answer with the correct one above
is a scalene triangle with perimeter 47; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47.
There are ten ways to add three distinct primes to yield sum 47:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:
The greatest possible length of the longest side is 23.
A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47.
There are ten ways to add three distinct primes to yield sum 47:
By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:
The greatest possible length of the longest side is 23.
Compare your answer with the correct one above