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If the perimeter of a right triangle is 12 and its area is 6, what is

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If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 02:56
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If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 03:40
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x
So sides can be 3,4, and 5 unit with perimeter 12 unit.
Area=1/2*3*4=6 sq unit

Smallest side=3

Ans. (B)
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 03:41
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6
Answer is B
Attachment:

1535542979079.jpg [ 76.21 KiB | Viewed 647 times ]

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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 04:19
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

As area is given, it could be figured out that there are only 3 combinations for various lengths of base and height.

12*1 = Not possible as the perimeter itself is 12

6*2 = Not possible because the sum of any two sides of a triangle must be more than the third side and 6 is half of 12 (perimeter)

3*4 = Possible. This is one combo of right angle triangle with sides 3, 4 and 5.

Thus the smallest side is 3.

Answer: B
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 06:04
PKN wrote:
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x
So sides can be 3,4, and 5 unit with perimeter 12 unit.
Area=1/2*3*4=6 sq unit

Smallest side=3

Ans. (B)

PKN
can you please elaborate on this Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x

what do you mean positive integer ? what is correlation betweeen positive integer and right triangle ? what it were negative integer ?
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 06:18
1
dave13 wrote:
PKN wrote:
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x
So sides can be 3,4, and 5 unit with perimeter 12 unit.
Area=1/2*3*4=6 sq unit

Smallest side=3

Ans. (B)

PKN
can you please elaborate on this Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x

what do you mean positive integer ? what is correlation betweeen positive integer and right triangle ? what it were negative integer ?
Hi Dave,

The side of a triangle can't be a negative integer. Imagine someone saying the side is - 4 cm. That is not sensical. I hope this helps. Feel free to ask a follow up question in case I was not clear.

Regards,
Sumit

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VP
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If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 06:21
hey there thank you but i stil dont get if side cant be negative integer

then why PKN emphasizes "perimeter are in the form of a +ve integer."
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 06:29
2
1
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

Matter of less than 10 secs if one is clear with the concept of Pythagorean Triples ( No need to remember, just go through this )
Attachment:

XfCMC.jpg [ 203.38 KiB | Viewed 566 times ]

https://www.mathsisfun.com/pythagorean_triples.html
So, its clear answer must be (B) 3
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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29 Aug 2018, 06:30
1
dave13 wrote:
PKN wrote:
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x
So sides can be 3,4, and 5 unit with perimeter 12 unit.
Area=1/2*3*4=6 sq unit

Smallest side=3

Ans. (B)

PKN
can you please elaborate on this Since area and perimeter are in the form of a +ve integer. Hence the sides of the right angled triangle are in the ratio 3x:4x:5x

what do you mean positive integer ? what is correlation betweeen positive integer and right triangle ? what it were negative integer ?

Hi dave13,

I was supposed to mention whole number but a whole number could be '0' too. So, I mentioned positive integer.

Why a positive integer?
1) Measure of side is always positive.side lengths of triangles cannot be negative
2) Perimeter is a positive whole number(a+b+c=12, NOT A DECIMAL) it means a,b,c are integers such as (3,4,5), (6,8,10) etc. Any of its sides can't be decimal number.
For example( You have to say if there is any triplet(a,b,c) which adds up to 12 and obeys pythagorean property)
Say (2.5,4.5,5) perimeter 12 but violates right angle property)

This is a quick observation.

Now, this quick observation rejects the possiblities of (a,b,c) being a decimal value. It directs us to classical Pythagorean triplets.
Now you know the answer.
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PKN

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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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03 Sep 2018, 18:04
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

We can let a and b be the legs of the right triangle and c be the hypotenuse of the right triangle. We can create the equations:

a + b + c = 12

½ab = 6

and

a^2 + b^2 = c^2

From the first equation, we have:

c = 12 - (a + b)

Substituting this in the third equation, we have:

a^2 + b^2 = [12 - (a + b)]^2

a^2 + b^2 = 144 - 24(a + b) + a^2 + 2ab + b^2

0 = 144 - 24(a + b) + 2ab

Since ½ab = 6, ab = 12 and 2ab = 24, we have:

0 = 144 - 24(a + b) + 24

24(a + b) = 168

a + b = 7

Substituting this into the first equation, we have:

7 + c = 12

c = 5

We see that the hypotenuse is 5, and the sum of the lengths of the two legs is 7. It’s not difficult to see that this must be a 3-4-5 right triangle. So the shortest side is 3.

Alternate Solution:

Let’s test each answer choice:

A) shortest side = 2

If the shortest side is 2 and the area is 6, then the remaining leg of the triangle must have a length of 6. Since the square root of 2^2 + 6^2 = 40 is not an integer, the perimeter of this triangle cannot be 12.

B) shortest side = 3

If the shortest side is 3 and the area is 6, then the remaining leg of the triangle must have a length of 4. Since the two legs of this right triangle are 3 and 4, we are dealing with a 3-4-5 right triangle and the perimeter is 12. This is the correct answer choice.

Answer: B
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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03 Sep 2018, 18:39
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

Let
'a' be the height,
'b' be the base,
'h' be the height

From question we know,

Area of Triangle -

$$\frac{1}{2} * b * a = 6$$

b * a = 12 ------ (1)

Perimeter of Triangle -

b + a + h = 12 -------- (2)

As its a right triangle -

$$b^2 + a^2 = h^2$$ ------- (3)

Now a general mathematical formula -

$$(b + a)^2 = b^2 + a^2 + 2ab$$ -------- (4)

Substituting (1), (2) and (3) in (4)

we get,

$$(12 - h)^2 = h^2 + 24$$

Expanding,
$$144 + h^2 - 24h = h^2 + 24$$
24h = 120
h=5 ------- (5)

From (5) and (2)

b + a = 7 -------- (6)

From (6) and (1)

Only possible integer value (as answer choices are integer only) is '3' and '4'.

Hence shortest side/value = 3

Answer = B
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is  [#permalink]

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04 Sep 2018, 05:28
Bunuel wrote:
If the perimeter of a right triangle is 12 and its area is 6, what is the length of the smallest side?

A. 2
B. 3
C. 4
D. 5
E. 6

a+b+c = 12
ab = 12
& a^2 + b^2 = c^2

multiple 2nd eq. by 2 and add to third eq.
=> a^2 + b^2 + 2ab = c^2 + 24
=> (a+b)^2 = 24 + c^2
=> (12-c)^2 = 24 + c^2
=> c = 5
now, finding a & b is not tough...the triplet is 3,4,5 so B.
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Re: If the perimeter of a right triangle is 12 and its area is 6, what is &nbs [#permalink] 04 Sep 2018, 05:28
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