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QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
Updated on: 01 Sep 2019, 02:39

2

Expert Reply

3

Bookmarks

Question Stats:

Two sides of a triangle have length 6 and 8. Which of the following can be the possible areas of the triangle?

Indicate all possible values

I. 2

II. 12

III. 24

_________________

Indicate all possible values

I. 2

II. 12

III. 24

_________________

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Originally posted by soumya1989 on 20 Jul 2016, 13:07.

Last edited by Carcass on 01 Sep 2019, 02:39, edited 4 times in total.

Last edited by Carcass on 01 Sep 2019, 02:39, edited 4 times in total.

Edited by Carcass

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
25 Jan 2017, 12:48

5

Expert Reply

soumya1989 wrote:

Two sides of a triangle have length 6 and 8. Which of the following can be the possible areas of the triangle?

I. 2

II. 12

III. 24

Indicate all possible values

I. 2

II. 12

III. 24

Indicate all possible values

Show: ::

I, II and III

As we can see from the diagrams above, all 3 areas are possible.

Area = (base)(height)/2

The area of triangle A = (8)(6)/2 = 24

The area of triangle B = (8)(3)/2 = 12

The area of triangle C = (8)(0.5)/2 = 2

Once we prove that, by positioning the two lengths at right angles, we can create a triangle with area 24, we can rotate the side with length 6 so that the triangle's height gets smaller and smaller, which allows us to create triangles with area 12 and 2.

Answer: I, II and III

RELATED VIDEO

_________________

Re: Two sides of a triangle have length 6 and 8
[#permalink]
20 Jul 2016, 19:32

3

The largest triangle will be right angle triangle with sides 6 and 8, so the maximum area will be 1/2*6*8 = 24. And the minimum area will be when one of the angle approaches 180 degree. When when of the angle approaches 180 degree, the area will approach 0. Hence all three choice are valid.

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
02 Jan 2017, 15:55

2

Hi-

so is the answer not B and C?

I thought that the third side must be at least 2+ (based on third side rule), if we multiply 2 *6/2, the smallest area I computed was 6.

Are we just supposed to assume that this third side is not the same as the height of the triangle? and we can make the area very small if the angle between the two larger sides is approaching zero?

Thanks,

Rahul

so is the answer not B and C?

I thought that the third side must be at least 2+ (based on third side rule), if we multiply 2 *6/2, the smallest area I computed was 6.

Are we just supposed to assume that this third side is not the same as the height of the triangle? and we can make the area very small if the angle between the two larger sides is approaching zero?

Thanks,

Rahul

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
06 Feb 2018, 20:28

1

Please, someone help me...

If two sides of a triangle have length 6 and 8 third side (x) has to between 2 < x < 14, which means answer A cannot be correct.

If two sides of a triangle have length 6 and 8 third side (x) has to between 2 < x < 14, which means answer A cannot be correct.

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
06 Feb 2018, 20:55

4

yell2012prime wrote:

Please, someone help me...

If two sides of a triangle have length 6 and 8 third side (x) has to between 2 < x < 14, which means answer A cannot be correct.

If two sides of a triangle have length 6 and 8 third side (x) has to between 2 < x < 14, which means answer A cannot be correct.

When the lengths of two sides of a triangle are given, there is no description on type of triangle you are dealing with. It may be a right triangle, isosceles or a very small triangle but not a straight line

and as such, you have to find the maximum and minimum value for the area

the maximum area will occur if the two sides form the legs of a right triangle.

Therefore Area \(= (\frac{1}{2})*(6)*(8) = 24 =\) maximum possible area

Now the range of possible areas is \(0 < Area\leq{24}\)

So all three values are possible.

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Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
Updated on: 25 Feb 2018, 10:34

1

Holy Cow! This problem is so devious! I also had B and C as my answers, but then realized that the height of the triangle can be chosen arbitrarily to make the area of the triangle equal to 2.

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
25 Feb 2018, 10:18

Expert Reply

Please avoid expressions such as Holy Christ. Please. For a stupid question, I do think is not necessary

Crap is already more than enough. Or similar to this.

Regards

_________________

Crap is already more than enough. Or similar to this.

Regards

_________________

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Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
25 Feb 2018, 10:24

Just relax there, dude. Not meant to offend or anything. Anyway, I've changed it.

Carcass wrote:

Please avoid expressions such as Holy Christ. Please. For a stupid question, I do think is not necessary

Crap is already more than enough. Or similar to this.

Regards

Crap is already more than enough. Or similar to this.

Regards

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
28 Aug 2019, 20:05

2

- Short Answer with no explanation

-. If you are given two sides of a triangle

. minimum area is practically 0

. maximum area is the area of right triangle formed by the two sides

. either side can be the base and the other side will be the height

-. Example

. Given triangle with sides of 6 and 8

.. minimum area = 0

.. maximum area = 24

.. therefore answer is A,B,C (or I, II, and III)

- Short Explanation:

-. Why?

. area is a function of height (1/2 base times height)

. height of a triangle is maximized when one of it’s angles is 90˚

. height is minimized when the angle is 0˚ or 180ˆ, at which point the area is basically 0

. so the range of area of any triangle formed from two given sides is at a minimum practically 0 and at a maximum the area of the right triangle formed by the two sides

See attached illustrations, or try out interactive triangle tool for yourself at: https://www.mathsisfun.com/geometry/pol ... ctive.html

- Long Explanation:

Basically here’s my thoughts on it

1) What is the maximum and minimum area possible? If we know that, then we can see if the answers are within that range

2) So, what’s the maximum area possible given the sides we were given?

. we know Area = 1/2 base times height

. but we don’t have height

. either 6 or 8 could be the base, but since we aren’t given any angles, we don’t know what the height is

. what possible height, given just these two sides, will give us maximum area?

. lets draw it out

. lets start with the lines basically flat

. what’s our height?

. it’s basically 0, maybe 0.0000000001

. so since our height is practically 0 our area would practically be 0

.. what if we start raising up the big side?

.. what happens to height?

.. we see it gets larger

.. as area is basically just the product of base and height we will have maximum area when we have maximum height

. so when will this happen?

. it’ll happen when the line is going straight up

. which is a 90 degree angle

. so then our area is 1/2 b • h = 1/2 6ª8 which is 24.

-. some might say, well, what about if our third side was the maximum possible, which is just the sum of six plus eight? Yes, then you would have a side of 14. But if you had that side, then the triangle would basically be flat

. and therefore your area would basically be 0.

-. so what about the opposite, what about when we have the rule for the third side which tells us the minimum length?

. well that would be slightly more than the difference which is 2. Well, if the third side is 2, what do we have? Again, you will basically have a closed triangle, which is flat.

. so the area would be practically 0

. so our maximum area, again, is when we have a right triangle because that’s when our height is the greatest. It doesn’t matter which side is the base, the result will be the same. You can tell just by looking at the formula, since we know we must use a right

triangle for maximum area then the base and the height in this case will just be the two given sides of the triangle, and they are interchangeable

- Note on attached illustrations

. I kept the base as 6 and the one side as 8

. I let the other side vary as I changed the angle between the base side of 6 and the 8 side

. if you look in the upper left corner you can see the area of the triangle and how it's maximized when the angle is 90˚

. and it's practically 0 when the angle is 0˚ or 180˚

-. If you are given two sides of a triangle

. minimum area is practically 0

. maximum area is the area of right triangle formed by the two sides

. either side can be the base and the other side will be the height

-. Example

. Given triangle with sides of 6 and 8

.. minimum area = 0

.. maximum area = 24

.. therefore answer is A,B,C (or I, II, and III)

- Short Explanation:

-. Why?

. area is a function of height (1/2 base times height)

. height of a triangle is maximized when one of it’s angles is 90˚

. height is minimized when the angle is 0˚ or 180ˆ, at which point the area is basically 0

. so the range of area of any triangle formed from two given sides is at a minimum practically 0 and at a maximum the area of the right triangle formed by the two sides

See attached illustrations, or try out interactive triangle tool for yourself at: https://www.mathsisfun.com/geometry/pol ... ctive.html

- Long Explanation:

Basically here’s my thoughts on it

1) What is the maximum and minimum area possible? If we know that, then we can see if the answers are within that range

2) So, what’s the maximum area possible given the sides we were given?

. we know Area = 1/2 base times height

. but we don’t have height

. either 6 or 8 could be the base, but since we aren’t given any angles, we don’t know what the height is

. what possible height, given just these two sides, will give us maximum area?

. lets draw it out

. lets start with the lines basically flat

. what’s our height?

. it’s basically 0, maybe 0.0000000001

. so since our height is practically 0 our area would practically be 0

.. what if we start raising up the big side?

.. what happens to height?

.. we see it gets larger

.. as area is basically just the product of base and height we will have maximum area when we have maximum height

. so when will this happen?

. it’ll happen when the line is going straight up

. which is a 90 degree angle

. so then our area is 1/2 b • h = 1/2 6ª8 which is 24.

-. some might say, well, what about if our third side was the maximum possible, which is just the sum of six plus eight? Yes, then you would have a side of 14. But if you had that side, then the triangle would basically be flat

. and therefore your area would basically be 0.

-. so what about the opposite, what about when we have the rule for the third side which tells us the minimum length?

. well that would be slightly more than the difference which is 2. Well, if the third side is 2, what do we have? Again, you will basically have a closed triangle, which is flat.

. so the area would be practically 0

. so our maximum area, again, is when we have a right triangle because that’s when our height is the greatest. It doesn’t matter which side is the base, the result will be the same. You can tell just by looking at the formula, since we know we must use a right

triangle for maximum area then the base and the height in this case will just be the two given sides of the triangle, and they are interchangeable

- Note on attached illustrations

. I kept the base as 6 and the one side as 8

. I let the other side vary as I changed the angle between the base side of 6 and the 8 side

. if you look in the upper left corner you can see the area of the triangle and how it's maximized when the angle is 90˚

. and it's practically 0 when the angle is 0˚ or 180˚

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Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
01 Sep 2019, 00:51

1

if there are only 2 unidentified sides,

we could put 2 or 3 values into A=1/2 b*h to test and

quickly assume the answers are all of the above and move on! XD

we could put 2 or 3 values into A=1/2 b*h to test and

quickly assume the answers are all of the above and move on! XD

Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
01 Sep 2019, 02:41

Expert Reply

[quote="arc601"][/quote]

Great explanation

_________________

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Great explanation

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Re: QOTD #6 Two sides of a triangle have length 6 and 8
[#permalink]
30 Jun 2021, 10:02

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