Not so Isosceles
A triangle with integral sides has two sides equal to 1 unit. What is the measure of the angle included between the two sides?
Give your answer in degrees.
The answer is 60.
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11 solutions
I disagree, if 2 sides are equal to 1 cm, then there is nothing pertaining to the problem that states that the third side can only be an integer? The only thing we know about the third side is that it must be less than or equal to 1. If the value is less than one, then the triangle would be isosceles, with many different solutions. In fact if you let the top angle equal "y", then x = (1/2)*(180-y). For example x = 75 would satisfy this solution.
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All sides are integral.
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It only says with integral sides, not that ALL are...
I guessed 65, 70, & 75 purely on the basis that the sum of the three angles of a triangle equate to 180deg. 60+60+60= 180. But seeing as there were only two defining marks of equality on the triangle, (both signing 1cm)I assume the third side must be less than 1. In turn the degrees must be less than 60 for the top angle Y. Naturally that which we take from Ang Y, we half, then add to Angs' X and Z. Even though I guessed by logic and deduction, with medial understanding of arithmetic, 60deg for Ang X seems so wrong.
You need to learn some maths
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He needs to pay more attention to the question thats all. He missed the "with integral sides "
If you cut the triangle into half, you can have a right triangle and be able to apply some of the trigonometric ideas. After you divided it into two, your hypotenuse is 1. then it has relationship with the unit circle. Then, it will be a 30-60-90 triangle. If we put it in the standard position, then the hypo is 1, short leg is 1/2 and longer leg is sqrt of 3/2. By the sine, you'll get, the sine of x is approximately, 0.866. What degree measure is equal or near to the sine value of 0.866 and cosine value of 0.5? Well, that's 60
Its a triangle with integral sides. Obviously the three sides are integers...
the question is not stated correctly....insufficient data. u cant judge any angle on the basis of two sides.
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Quite right. The problem as stated is unsolvable.
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However, you can get the third side from the problem. Since two sides of a triangle always add up to more than the third side, the third side would have to be one, since the problem states that they have integral sides. So, since all sides are 1, it is an equalateral triangle, and angles are 60
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@Gaurav Rao – It is not clear from the English language that ALL are integers.
However, you can get the third side from the problem. Since two sides of a triangle always add up to more than the third side, the third side would have to be one, since the problem states that they have integral sides. So, since all sides are 1, it is an equalateral triangle, and angles are 60
You are absolutely right ..
Awesome answer .
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Not Valid Solution....Third side might be .1 to .9
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Even if you did not notice "integral", why on earth it should be .1 to .9?? It could be anything between 0 and 2( except these two)
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@Bikash Kumar – So angle x is between 0 and 90 degrees, exclusive.
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@Jojo Geronimo – yes you are right. angle can be anything from 0 to 90 except these 2.
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@Bikash Kumar – My friend please check the meaning of integral again.
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@Ansh Bhatt – according to you what is the meaning of integral?
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@Bikash Kumar – Integral means being an integer. This isn't what I say. Google and Oxford dictionary say so.
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@Ansh Bhatt – If you believe I know what integral means. My last comment was regarding an answer submitted assuming integral does not mean integers. Even if integral does not mean integers. third side can be anything from 0 to 2 (except these 2). But some of the friends above were saying it should be between 0.1 to 0.9 I just wanted to make clear why should the value of third side be between .1 to .9
@Bikash Kumar – Because the word integral does not mean " are integers" ...just means" are a necessary part" of. Also does not describe ALL as integers!
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@Carol Blyth – I agree Integral does not men integers but tell me why should it be between .1 and .9? If 2 sides are 1 and corresponding angles are equal. third side can be anything from 0 to 2 (except these 2)
think again,
sum of any two sides must be greater than the third side.
sum of the given two sides is 2. hence the third side must be smaller than 2. as it was given that measure of third side is integer the no. must be either 1 or 0 . it can't be zero hence answer is 1
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@Chaitanya Lodha – Also, the sides are integers.
the problem says integral sides. no number between 0.1 & 0.9 is an integer
Sides must be integers
Read the question. Its said, sides are integral in length
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@Suhas Heeraskar – How many sides? The sides are integers YES. All sides are integers YES Has integral sides NO. That can mean 2 or 3 sides, so a badly written question with no answer.
That's a good answer but the question to me is badly written. 'Integral' sides does not imply that the sides are integers (whatever that would mean anyway). Integral sides just mean they are part of the triangle or some such.
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Sometimes Integral is used instead of Integer. I agree Integer is better.
could anyone please tell me how this image got attached with my problem? (it looks better now, but the question arose as i did not attach any image)
Just loved your way of presenting !!
The angle can be above 60 degree and less than 90 degree
Good work I like it
it does not need to be necessarily an integer..
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Integral means that the lengths and area of a shape are both integers so it must be an integer.
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If the third side is 1 cm, then the area of the triangle is not an integer anymore. =)
If the angle is 60 then that is a very very wrong diagram!!! Think about it again! :)
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Diagrams tend not to be to scale in most tests anyway :P Although they should have written here diagram not to scale because that is always noted in exam papers
third side might be some integer with ' mm' unit like 50mm, ,,,,am i write ?
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I agree. Metric system has units that are multiples of ten. 1 cm = 10mm If you rewrite the diagram in mm the you have side 10mm, 10mm and (1-9mm) meaning the third side does not have to be a 1 or 0 to be an integer.
No. the question says that the triangle has an integral side. Hope your doubt is clear now. :)
I understand that the third side length must be an integer from "integral", however I am confused about your statement "The sum of 2 sides must be greater than than the third side." without that statement the third side could be any integer and the angle could be any up to and including 178
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It is a general rule of triangle. That the sum of any two sides will always be greater than 3rd side.
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you're telling me that that in an obtuse isosceles triangle the sum of the legs is greater than the base?
h= Sqrt(b^2 - (a^2/4))
60 good one
If each section measures 1 cm & divided the triangle in two triangled then cos x=0.5/2; x=75° Each angle shown is 75°
The included angle then is 180-2 (75)=30
I see there has been much discussion here, I certainly don't mean to re-hash any of the previous comments. I instinctively knew the answer, but was confused by the statement "two sides equal to one unit." Approaching this as a logic problem, I assumed only two sides were equal to one unit. I realized what you were asking after I viewed the solution. BTW, I fully admit the problem in understanding lies with me and not the original problem.
cannot be, triangle must have atleast 3 known values to solve the problem . . .
sorry, the side could also 0.2 - 0.9
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side length must be integer. it is mention in problem
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This answer is valid only if it were mentioned as 1 "unit" and not 1 "cm" because even 10mm or less than that can be considered as "integral values" maintaining the fact that it is less than 1 "cm". Hence there will be multiple values for this problem.
Let the length of the third side be denoted by x
Since the lengths of sides are integers
and the sum of lengths of any two sides is greater than the length of the third side
Then
1 + 1 > x
x <2
x = 1
So the triangle is equilateral
Then , the measure of each angle is 60
Where does it say that all 3 sides must be integers? I agree that the 3rd side must be less than the sum of the other 2, but I don't see how you are getting it to be 1. It is simple enough to represent an equilateral triangle graphically, I don't understand why they would scale the triangle differently, if it was meant to be equilateral. It looks nothing like an equilateral triangle. The two bottom angles look much larger than the top one.
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yes, I agree with Marcus Worrell because it might not rely on the given condition. And, also, it might confuse the reader because the given and the figure are not on the same track.
The answer you gave given is wrong because the length of third side can be any number between 0.1 to 0.9
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They said "integral sides" so the lenght must be an integer, so x <2 means x=1 or x=0, but it's a triangle, so x=1
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This answer is valid only if it were mentioned as 1 "unit" and not 1 "cm" because even 9mm or less than that can be considered as "integral values" maintaining the fact that it is less than 1 "cm". Hence there will be multiple values for this problem.
Integral sides mean the side length can only be an integer. According a rule about triangle that one side cannot be more than the sum if the other two sides. In this case, the third side cannot be nor be more than 2. It also cannot be 0; hence, it is equal to 1 since that is the only integer between 0 and 2. Since all sides are equaled to 1, it is an equilateral triangle. The angle is 60° according to the definition of equilateral triangles. QED
let the triangle be ABC with angles B =C . Let the third side be x then cosB = 2 ∗ 1 ∗ x x 2 + 1 − 1 = 1/2 Therefore B =C=A =60 Therefore third side is 1
Look at the regular sin graph. You can make and educated estimate quiute easily.
Guys, this is just a general message. Please at least look into a dictionary before making any comments. All those who feel the word integral is used wrongly please update your dictionaries.
This is a terrible question. It lacks vital information about the third side, leaving solvers to make assumptions about it. Only knowing the length of the two equal sides tells nothing about the third side. The two equal sides of the triangle will of course each subtend the same angle on the third side, but this could be anything from 1 degree or below, to almost 90...of course it could not be 90...that would make it a line. I made the assumption from the drawing was that the base was half the length of the sides. I then bisected this line and used trigonometry to work out that the angle in question was cos .866. That is 60.
If the two sides are 1cm each, the third side must be less than 2cm. Since the sides are integral, and the only integer between 0 and 2 is 1, then the triangle is equilateral, thus all three angles are 60 degrees.
here is another solution too :)
If a triangle has two sides of length 1, and we know that the triangle has all integer side lengths, the only possibility will be 1. This is because of the triangle inequality which states that the sum of the two smallest sides of a triangle must be greater than the largest side.
If we look at the possibilities for the last side, we know that we have two possibilities 1 or 2 because anything less than or equal to 0 is impossible; you can't have a negative length or a 2 sided figure. When we look at 1,1, 2 we realize that 1+1 is equal to 2, not greater. This means that this is an invalid pair. So the only possibility is 1, 1, 1 which does work for the triangle inequality.
Seeing that all three sides are equal, we know that it is an equilateral triangle and that the answer is 60 degrees.
let the other side be x. Then by triangle inequality, 1+1>x (sum of 2 sides of triangle is greater than the third side) 1-1<x (difference of 2 sides is smaller than the third side) therefore 0<x<2 since x is an integer therefore x=1 all sides of triangle equal to 1 therefore equilateral triangle all angles of equilateral triangle is 60 deg
The sides are integral.
The sum of 2 sides must be greater than than the third side.
Given 2 sides 1cm and 1cm so their sum 2cm must be greater than the third side.
The only interger satisfying this is natural number less than 2, that is 1cm.
So, the triangle is equilateral and angle between any two side is 180/3 = 60 degrees.