In triangle PQR, if the length of all the three sides are distinct int

Question

In triangle PQR, if the length of all the three sides are distinct integers, what is the area of triangle PQR?

I.  PQ^2 = PR^2 + RQ^2 
II. The perimeter of triangle PQR is equal to 12 units.

GMATWhiz Quant Prepathon Lesson 2 Question 1

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GMATWhizTeam · Answer

I hope you were able to internalize the second skill and understand how to consider all cases. Here's the detailed solution to the first question. Even if you got the question right make sure that you check out the method in the solution to ensure that you have solved it using the right method. All the best!

Feel free to post any doubts that you may have in the forum below. We will be happy to help you with your queries

GMATWhizTeam · Answer

Sal91   Sappy   Nikhil099   preetamsaha  Here's the first question for Lesson 2. Do take a shot at it and share your explanations in the comments. All the best!

minustark · Answer

In triangle PQR, if the length of all the three sides are distinct integers, what is the area of triangle PQR?

I.  PQ^2 = PR^2 + RQ^2 
II. The perimeter of triangle PQR is equal to 12 units.

1) Since all three sides are distinct integer, so we can infer this is a right triangle.
2) 3 sides can only be 3, 4 and 5. So this is a right triangle. We can get the area from it. Sufficient.
B is the answer.

GMATinsight · Answer

The video explanation of the problem can be seen here.

https://www.youtube.com/watch?v=fQMy3tW98kQ

shameekv1989 · Answer

In triangle PQR, if the length of all the three sides are distinct integers, what is the area of triangle PQR?

I.  PQ^2 = PR^2 + RQ^2 
II. The perimeter of triangle PQR is equal to 12 units.

Given :- a,b,c (sides of the triangle) are distinct integers
To find :- Area of the triangle

i) We can just infer that the triangle is right-angled - Insufficient

ii) Perimeter = a+b+c = 12
Now we know, 3+4+5 = 12 => A right angled triangle is definitely one of the cases.

To try out others, we need to make sure that the "Sum of any 2 sides of the triangle is greater the third side and Difference of any 2 sides is less than the third side"

Let's try with the shortest side

Let's start with a=1 - Whenever we have a side as 1 it will be the shortest side and the difference of other 2 sides can not be less than 1 since the other 2 sides can not be equal (when the diff would be 0; else difference would  geq{1} ). For eg. 
a = 1; b=2; c=9 => c-b = 7 > 1
a = 1; b=3; c=8 => c-b = 5 > 1

When a = 2; b + c = 10 but b and c have to be consecutive so that the difference is 1 and not  geq{2}  i.e. the third side
a = 2; b=3; c=7 => c-b = 4 > 2 (b and c are not consecutive and we can see why this can not form a triangle) 
in fact b and c can not be consecutive and add upto 10

When a=3; b+c = 9 - When the shortest side is 3, we can not have any sum other than 3,4,5 values that will add upto 12 and have distinct integers at the same time. 
a=3; b=4; c=5 
Sufficient

Answer - B

preetamsaha · Answer

In triangle PQR, if the length of all the three sides are distinct integers, what is the area of triangle PQR?

Given :- a,b,c (sides of the triangle) are distinct integers
To find :- Area of the triangle

i) We can just infer that the triangle is right-angled - Insufficient

ii) Perimeter = a+b+c = 12
Now we know, 3+4+5 = 12 => A right angled triangle is definitely one of the cases.

To try out others, we need to make sure that the "Sum of any 2 sides of the triangle is greater the third side and Difference of any 2 sides is less than the third side"

Let's try with the shortest side

Let's start with a=1 - Whenever we have a side as 1 it will be the shortest side and the difference of other 2 sides can not be less than 1 since the other 2 sides can not be equal (when the diff would be 0; else difference would $geq{1}$). For eg.
a = 1; b=2; c=9 => c-b = 7 > 1
a = 1; b=3; c=8 => c-b = 5 > 1

When a = 2; b + c = 10 but b and c have to be consecutive so that the difference is 1 and not $geq{2}$ i.e. the third side
a = 2; b=3; c=7 => c-b = 4 > 2 (b and c are not consecutive and we can see why this can not form a triangle)
in fact b and c can not be consecutive and add upto 10

When a=3; b+c = 9 - When the shortest side is 3, we can not have any sum other than 3,4,5 values that will add upto 12 and have distinct integers at the same time.
a=3; b=4; c=5
Sufficient

Answer - B

sauravleo123 · Answer

Given triangle PQR
All 3 sides of the triangle are distinct intrgers.
Area triangle PQR=?

Statement 1: PQ^2 =PR^2+RQ^2
PQR is a right triangle
We have a infinite numbers of right triangles with sides having integers value.
Insufficient
Statement 2: PQ+PR+RQ=12
We know that the sum of addition of 2 sides of a triangle must be greater than the third side.
The 3 sides have to be distinct integers as well.
3,4,5 is a possibility
2,5,5- not distinct
2,3,7- violates the triangle rule
Whatever combinations we try, we see that 3,4,5 is the only possibility.
We have the values of the 3 sides of a triangle.
Area can be calculated.
Sufficient
B

GMATWhizTeam · Answer

The official solution is shared. You can find it in the first post right below the question. Enjoy learning and ensure that you participate in the other questions of the Prepathon if you missed this one.

In triangle PQR, if the length of all the three sides are distinct int

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