What Is The Length Of The Square's Side?

A triangle with sides of 4, 3, and 5 is drawn inside a square as shown. What is the length of the square's side? It's a fun little geometry problem!

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[asktnegi]

Presh, you stopped asking DID YOU FIGURE IT OUT and that's the only thing which makes your channel the best. Please kindly start asking that because that triggers a challenge mode inside us.

[pauldzim]

For Presh’s geometry problems, 90% of the time it’s similar triangles

[XJWill]

Interestingly enough, this problem can still be solved even if the inscribed triangle is not a right triangle. But since the other three triangles all have a corner of a square, the other three triangles are all right triangles even if the inscribed triangle is not. So by using the Pythagorean theorem on those three triangles together with the fact that some of the legs of the triangles combine to form the edge of the square, it is possible to solve for the square's size. It is just algebra, but it gets quite tedious since there are a lot of square roots to clear by squaring both sides of the equations and squaring obviously adds the possibility of superfluous spurious solutions. But it does work -- I checked it by solving the given problem without using similar triangles.

[vidhisingh]

I applied trigonometry in it.
Sin a= ed/4
Ed = 4 sin a
Cos a= dc/4
Dc = 4 cos a
And in upper triangle cos a= ae/3
Ae=3cos a
But ae +ed =dc
3cosa +4sina = 4cos a
4sin a = cos a
Tan a = 1/4
If we evaluate cosa = 4/√17
DC = 4 cosa = 16√17 and that's the side of a square

[TMadhusudan]

You have stopped saying, " did you figure it out ?". ☹️

[mitchblank]

I think the problem can be improved by just asking for the ratio of the triangles area to the square's. Then you get a nice rational 51/128 at the end, and you can send people down some more blind alleys on the way.


If you want to generalize to any interior right triangle with lengths a, b (i.e. 3, 4 in this example) you get the gnarly ratio (a³+2ab²-2a²b)/2b³

[dwaynemadsen]

Thank you for sharing! I like watching these every few days so I am resisting the urge to binge watch your entire library! Stay safe.

[احمدمحمد-تطج]

تمرين جميل جيد. رسم واضح مرتب.واضح مرتب. شكرا جزيلا لكم والله يحفظكم ويحميكم ويرعاكم. تحياتنا لكم من غزة فلسطين

[JerusalemStrayCat]

I love these kinds of problems because you can approach the problem from all sides except one and still not get it. There was ONE missing piece I was trying to find, and once I got to that part of the video it all fell into place

[ghhdcdvv]

تمرين جيد . رسم واضح مرتب . شرح جيد واضح .شكرا جزيلا لكم والله يحفظكم ويحميكم ويرعاكم . تحياتنا لكم من غزة غلسطين .

[jackbrady]

wow so well explained. If i was taught how to solve "problem" solving questions like this when I was still learning maths in high school I would've enjoyed it thoroughly. thanks Presh

[Immortal]

Had it been before I started watching your videos, I wouldn't have got this one.
However, I watched enough to look for similar triangles, and was able to solve this.

[nooralam]

Sir, my brother trying to solve by triangle ( similar angle) but don't complete solved..
First time, I win by this question..
I am very happy 😀😀😀
Nice question

[vacuumcarexpo]

I barely managed to get the answer solving a fourth degree equation. I didn't know such an easy way. Thanks a lot.

[carrollstewart]

the triangle on right side of the square is a right triangle with Hypotenuse of 5. That makes the top of the triangle 3. The other side of the triangle is 4 and since it is the entire side of the square, then x also =4

[PuzzleAdda]

You learn lot of things from this YouTube channel. This youtube channel is like god father for my channel.

[akio-the-lazzycatto]

If we calculate the square area without a triangle inside and multiply it by 2, we get the square area + sqrt((25 - x^2)(16 - x^2), then let t = x^2
17t = 256
x = 16/sqrt(17)

[justinlouisantonio]

I thought the length would be 4, because the 5 made another right triangle in other side.. but the 4 in the first triangle is the 4²=a²+b² which the length can't be equal to 4. I'M CONFUSED

[maruthasalamoorthiviswanat]

Beautiful Explanation. I could not solve it sir. THANK YOU SIR

[sashankbabu]

One small doubt..? If you take right side triangle located on the square ...it looks like, x^2 + [x-{x/4}]^2 = 25 //:as (y= x/4) by simplification, x^2= 16 and x is 4 ..! I'm just a student, but I would like to know, am I right in this ..? Please let me know Sir, I'll be waiting for your reply..!