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Can anyone help with this geometry question?


find the value of x?

I was able to find the answer given (12.8) by assuming that the triangle with the x variable is a right triangle, but I can’t prove it. is this a right triangle? and if so, then please say how?


Hello Harisson!

It’s really challenging and I have graduated from high school before twenty three years, so I don’t remember that much. Anyhow, I tried many times to think how it could be solved and maybe without following a good idea but maybe a paradox one. A right triangle you mean an equilateral one? Because it’s not and neither an isoskeles. It looks like more an obtuseanged or an oblique, which means that every side has different distributions that can be measured with precision and with a protractor; If every triangle has 180° then there is a way to find X. But anyway if you have found how to solve it then send me a message because I’m interested to know how you have found 12.8. Is that the correct answer?
Also, right triangle you mean that it’s bigger but equal to the smaller in regard the 180°, because they don’t look similar to each other. In addition, I believe that inside the circle there are two fives, from O to the circle followed by the blue line is again 5 but how much is the rest of it. And for example if 8 is 54° and five was 40° how much would be the rest? 180-94? I know it’s crazy but can we measure with a protractor and find a clue? And how about the Pythagorean theorem! can be applied here? X is the longest side of the triangle and somehow the same side on the smaller triangle looks alike. If a2+b2=c2 then 25+64=89 the root of 89 is 9,4340
But how we can find in the same way the sides of the bigger triangle! The funny is that if you have the right number for x then we must find the third side :slight_smile:
I believe that it isn’t 12.8 because this is the number I get if x is 12.8 as you said for the third side. Why I don’t believe it? Because x looks longer than the other side next to the circle.
I think that x is 13,4 or something close but I’m not sure. The side next to the circle is almost 9 so 9×9=81. 10×10=100. 100+81=181 so what is the root of 181=13,4536
It’s only my opinion and I hope I won’t confuse you more.



12.8 is the answer in the book, I will avoid calling it the correct answer until I can find a way to calculate it without making an assumption. The method I use to get this answer involves assuming that the bigger triangle is a right triangle (i.e. a triangle with a 90degree angle in it). This is important because Pythagoras theorem only applies to right triangles.

to get 12.8, I noticed that the 2 tangents to the circle have a common point. The tangents are the side with length 8 and the unlabelled side of the bigger triangle. (i.e. the bigger triangle has a side labelled length 10, and another x, then the third is unlabelled, I will call it y)
the side with length 8 equals the length of side y because they are tangent segments that extend from the same point.
If the bigger triangle is a right triangle, we can find x using the Pythagoras theorem, 10^2+8^2 =164
square root(164) = 12.8

I think you highlighted the exact problem with assuming that the bigger triangle is a right triangle. I say it is a right triangle and you say it looks more like an obtuse triangle and someone else might call it isosceles or equilateral. Unless someone can provide a reason that is more precise than that’s what it looks like we cannot be sure which one is correct. The drawings are also not to scale so going by looks is unreliable. this is also why we cannot use a protractor/ruler to just measure the sides.


Then it has to be correct if the answer is 12.8, it makes sense the third side to be 8, and if the book gives it. Also it looks from the shape that X is the bigger one and the third united with the circle the smallest.
As long I remember if you magnify any triangle (make the shape bigger) they still have the same degrees. The thing that I don’t get is that in the other triangle the side which is 8cm looks smaller than the unknown in the bigger triangle, which means that designer didn’t care of make it look more accurate. Just assumptions making, and as you said it’s impossible to use the theorem for isoskeles and equilateral triangles because some of them have 60° degrees each side in order to give 180°. Thanks anyway, I will have free time at weekend to study this more.


Yes, the answer is approximately 12.8. Exactly, its sqrt(8^2+ 10^2)

If you draw in the missing radius, which is 5, and use the fact that tangent lines to circles are always perpendicular (forming right angles), you see that you have 2 small congruent right triangles next to each other. The 2 congruent triangles share a hypotenuse of unknown length, but you don’t need it. The 2 legs in both small triangles are 5 and 8.

Therefore, the large right triangle has legs 8 and 10.
To find x, the hypotenuse, use the Pythagorean Theorem. x^2 = 8^2 + 10^2 :relaxed:


sqrt(8^2 + 10^2), so approximately 12.8, is the correct answer assuming that the triangle with a leg labeled x is a right triangle. Without that assumption, however, x can be any number between approximately 1.99 and 20, but the vast majority of such triangles will not resemble a right triangle. The large triangle looks like a right triangle, but that it is one cannot be proven without further information,