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QillerDaemon wrote: ↑Sun Apr 02, 2023 2:38 pm Good try, but unfortunately wrong. The answer is an integer with three places, and over three times as big as your answer. While there are other ways to solve the problem, the two tangent theorem makes it easy. Problems like this, unless they involve solutions on circular diagrams, are almost always integer answers, and even if circular, almost involve integer multiples of pi or easy fractions of pi.

The two tangent theorem states that two line segments tangent to the same circle that meet at a point outside the circle have the same length from the tangent point to the outside point, ie congruent. That helps you set up formulas for the lengths of the unknown sides of the triangle. Pythagorean theorem and Bob's your uncle.

QillerDaemon wrote: ↑Sun Apr 02, 2023 2:38 pm Good try, but unfortunately wrong. The answer is an integer with three places, and over three times as big as your answer. While there are other ways to solve the problem, the two tangent theorem makes it easy. Problems like this, unless they involve solutions on circular diagrams, are almost always integer answers, and even if circular, almost involve integer multiples of pi or easy fractions of pi.

The two tangent theorem states that two line segments tangent to the same circle that meet at a point outside the circle have the same length from the tangent point to the outside point, ie congruent. That helps you set up formulas for the lengths of the unknown sides of the triangle. Pythagorean theorem and Bob's your uncle.

Bluespruce1964 wrote: ↑Wed Apr 05, 2023 11:58 am

QillerDaemon wrote: ↑Wed Apr 05, 2023 5:51 pm The correct answer, good buddy! The hint was a suggestion to use for a geometric solution, but you used a(n equally valid) trigonometric solution. Whatever gets the answer. The two tangent theorem (TTT) is useful at odd times. Imagine a dunce cap on CTC's round idiot noggin. No matter how you place the cap on his dome, the two lengths from where the cap touches his coconut to the tip of the cap are of equal length.

Put the triangle so the longer length sits on the base and the shorter length is to the left. Draw the tangent circle to all three sides. Drop a segment from the circle's center down to the base leg, and another segment from the center to the left leg. With the triangle's right angle and these two lengths, it makes a square of 12 units on each side. More important, it shows that the left leg is made of two sub-segments of 12 units and (32 - 12 =) 20 units. It also shows that the base leg is also made up of two sub-segments of 12 units and an unknown length X. These sub-segments are divided where the circle touches each leg of the triangle. And lastly, by the TTT, the hypotenuse is made up of two sub-segments of 20 units and some unknown length. The 20 unit length passes from the left leg, around the crown of the circle, and over to the hypotenuse.

Going to the point where the hypotenuse meets the base leg, the TTT again shows that the two unknown lengths are exactly the same, X. So now we know all we need to set up the Pythagorean Theorem (PT). The left leg length is 32 (given), the base leg is (X + 12), and the hypotenuse is (X + 20). Putting these values into the PT --> 32² + (X + 12)² = (X + 20)², Expand, drop, and reduce, this settles down to X = 48 units. So we now know that the left leg is 32 units, the base leg is (48 + 12 =) 60 units, and we don't need to know that the hypotenuse is (48 + 20 =) 68 units.

From the formula for the area of a triangle, A = 1/2(B)(H), B = 60 units, H = 32 units, and solving, A = 960 units².

Animal wrote: ↑Wed Apr 05, 2023 6:45 pm
Honestly, I don't ever remember that formula,

This problem did bring up an interesting thought, though. Obviously this triangle is unique in that all three sides are integers. 32, 60 and 60. The most famous of those triangles is the 3,4, 5.

So, is there a formula to determine the size of a circle that is tangent to all three sides of a right triangle?

QillerDaemon wrote: ↑Thu Apr 06, 2023 3:11 am

Animal wrote: ↑Wed Apr 05, 2023 6:45 pm
Honestly, I don't ever remember that formula,

This problem did bring up an interesting thought, though. Obviously this triangle is unique in that all three sides are integers. 32, 60 and 60. The most famous of those triangles is the 3,4, 5.

So, is there a formula to determine the size of a circle that is tangent to all three sides of a right triangle?

Yea, it's one of those tools you forget about, stuck in the bottom of your tool chest, with all sorts of more usable tools piled on top of it. And then, you come across a problem and that's just the tool to fix it. Since you used trig, you didn't need it.

Those are Pythagorean triplets. The 3-4-5 is the simplest one, so is 5-12-13. And the triangle in the original problem is also a triple when reduced down to lowest terms -> 8-15-17 (32/4, 60/4, 68/4). It'd be an interesting question whether any triangle made up of a Pythagorean triple has an inscribed circle whose radius is also an integer.

And yes, there is such a formula for that. The radius of an inscribed circle tangent to all sides is (1/2)[(side A) + (side B) - (hypotenuse)]. That's based on Heron's formula, and I had to Google all that. For the triangle in the original problem, (1/2)[32 + 60 - 68] = (1/2)[24] = 12.

So for 3-4-5, the inscribed radius is 1, for 5-12-13, it's 2, and for 7-24-25, it's 3. Also notice with all of those triples, the base and the hypotenuse are often so close together in value, it kind of reminds of how sin(θ) ≈ θ (in radians) for values of θ < 14°.

QillerDaemon wrote: ↑Thu Apr 06, 2023 3:11 amThe radius of an inscribed circle tangent to all sides is (1/2)[(side A) + (side B) - (hypotenuse)].

disco.moon wrote: ↑Thu Apr 06, 2023 10:08 am I like to check in this thread to remind myself how stupid I am. ...

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Ahhhh that'll do it.

Animal wrote: ↑Thu Apr 06, 2023 12:47 pm So, if a triangle has integer sides (3-4-5 or 5-12-13) then the inscribed circle has a radius that is an integer? Is that true in every case?

QillerDaemon wrote: ↑Thu Apr 06, 2023 3:56 pm

Animal wrote: ↑Thu Apr 06, 2023 12:47 pm So, if a triangle has integer sides (3-4-5 or 5-12-13) then the inscribed circle has a radius that is an integer? Is that true in every case?

I tried about twenty triples, and they all produced an integer radius. But that's not a proof. And mathematical proofs were never a strong point for me. Only takes one negative case to disprove the whole notion, but maybe a proof by contradiction. Like D.M, I'm just not that smart...

Another formula for finding the radius of an inscribed triangle, not necessarily a right triangle, also comes ultimately from Heron's formula: if A is the area of the triangle, and P is its perimeter, then the radius of the inscribed circle is R=2A/P. In fact, all those formulas come from Heron's formula, just rearranged in various ways for various goals. So I think what you asked about in your second reply is true, just working backwards.

disco.moon wrote: ↑Thu Apr 06, 2023 10:08 am I like to check in this thread to remind myself how stupid I am. ...

QillerDaemon wrote: ↑Thu Apr 06, 2023 4:19 pm

disco.moon wrote: ↑Thu Apr 06, 2023 10:08 am I like to check in this thread to remind myself how stupid I am. ...

Nah, not at all. As Animal mentioned, math nerds are their own species. Back in the mid 80's I wanted to go into computer science. But at the time, and probably still true, they did not allow major transfers, and I was an English major. Either you started out in the CS dept as a freshman, or you didn't and had to find another major. But the Math dept had a "back door" CS degree: BS in Math with an automatic minor in CS. Basically a CS degree heavy in math. I'd have to take a base set of math classes, but would be allowed take practically any CS class I wanted. That lasted about a year, not due to the CS classes but due to the upper level math classes. At that level, it's almost never about formulas and finding answers, it's all about mathematical proofs to various propositions. Logic and lots of it! I hung around a few math nerds, and it was very discouraging, they were like in a different thought universe. My mind didn't work like that, and I simply couldn't make it. Got all A's in the CS classes, and almost failed at the math classes.

Take topology, the math of shapes and their properties and relationships. That's the math that takes a donut and reshapes it into a coffee cup. Pretty esoteric, huh? In a typical topology text, that covered on page 2. The rest of the 475 pages? Might as well be reading a wizard's spell book.

After a year, I changed over to biochemistry with a chemistry minor, and never looked back, except when it came to physical chemistry, which I did rather poorly in. But biochemistry was a real blast, it was just at the cusp of "quick" DNA testing (which still took weeks and a large clean specimen), and we couldn't help but wonder where it was leading to. My own interest was biochemical evolution, how the chemistry of life formed. Really interesting stuff!

QillerDaemon wrote: ↑Thu Apr 06, 2023 9:16 pm

Animal wrote: ↑Thu Apr 06, 2023 4:01 pm damn, that's a simple formula. R=2A/P.

For your example, the sides were 32, 60, 68. R=2(960)/160 = 12. So that works.

I don't know why that is kind of mind blowing to me, that all of these integer triangles also have integer radiuses on the inscribed triangles. yet, i don't see a very simple proof to it.

QillerDaemon wrote: ↑Wed Apr 05, 2023 5:51 pm Imagine a dunce cap on CTC's round idiot noggin. No matter how you place the cap on his dome, the two lengths from where the cap touches his coconut to the tip of the cap are of equal.

rule34 wrote: ↑Sat Apr 08, 2023 5:34 pm If you lack $14.00 having 47 cents how much money do you have?