Simple geometry problem -- Find the perimeter of the triangle ABC

In summary: Yeah, I just answered quickly without reading the thread. A better explanation was also provided proving the equal triangles.
  • #1

docnet

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Homework Statement
what is the perimeter of the triangle ABC?
Relevant Equations
##\angle A+\angle B + \angle C = \pi##
Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it.
Screen Shot 2021-10-11 at 12.28.39 PM.png

what I know:
The internal angles add up to pi
Let the tangent point between A and B be X
Let the tangent point between B and C be Y
Let the tangent point between C and A be Z
## \overline {AC} = 14##
## \overline {BX} = 16##Things I suspect are true, but have yet to prove:
##\overline {AX} =\overline {AZ}##
##\overline {CZ} =\overline {CY}##
##\overline {BX} =\overline {BY}##

Relations:
##\overline {CY}=14-\overline {AZ}##
##\overline {AX}=\overline {AB}-16##so far I only found three unknowns and only two equations! I think that this is strangely difficult.
 
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  • #2
Um, the circle is circumcised?

...anyway, yes, these are true:
docnet said:
Things I suspect are true, but have yet to prove:
##\overline {AX} =\overline {AZ}##
##\overline {CZ} =\overline {CY}##
##\overline {BX} =\overline {BY}##
which you can see by, e.g. considering the right angled triangles ##AXP## and ##AZP## with common hypotenuse, etc.

Now you can see the answer just by looking at the figure, but if you like, denote the three relevant lengths by ##a,b,c## (i.e. ##a+c = 14, b=16##) and do it algebraically.
 
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  • #3
@ergo: you rob docnet of the exercise this way!
PF normally restricts help to guiding questions and hints...
 
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  • #4
ergospherical said:
Um, the circle is circumcised?

...anyway, yes, these are true:

which you can see by, e.g. considering the right angled triangles AXP and AZP with common hypotenuse, etc.

Now you can see the answer just by looking at the figure, but if you like, denote the three relevant lengths by a,b,c (i.e. a+c=14,b=16) and do it algebraically.
aha! the perimeter of the triangle is 16+ 16+14+14 =60. thank you ! it seems obvious now

and I did not realize that it says circumcised. It was probably a typo.
 
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  • #5
ergospherical said:
the circle is circumcised?
Else the triangle would have been larger.
 
  • #6
docnet said:
Homework Statement:: what is the perimeter of the triangle ABC?
Relevant Equations:: ##\angle A+\angle B + \angle C = \pi##

Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it.
View attachment 290564
what I know:
The internal angles add up to pi
Let the tangent point between A and B be X
Let the tangent point between B and C be Y
Let the tangent point between C and A be Z
## \overline {AC} = 14##
## \overline {BX} = 16##Things I suspect are true, but have yet to prove:
##\overline {AX} =\overline {AZ}##
##\overline {CZ} =\overline {CY}##
##\overline {BX} =\overline {BY}##

Relations:
##\overline {CY}=14-\overline {AZ}##
##\overline {AX}=\overline {AB}-16##so far I only found three unknowns and only two equations! I think that this is strangely difficult.
Notice the three isoscelese "triangles" created by the circles tangent points? You don't need any trigonometry to solve this only basic algebra and that fact.
 
  • #7
valenumr said:
Notice the three isoscelese "triangles" created by the circles tangent points? You don't need any trigonometry to solve this only basic algebra and that fact.
As noted in post #2, and used in post #3.
 
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  • #8
haruspex said:
As noted in post #2, and used in post #3.
Yeah, I just answered quickly without reading the thread. A better explanation was also provided proving the equal triangles.
 

1. How do you find the perimeter of a triangle?

The perimeter of a triangle is the sum of the lengths of all three sides. To find the perimeter, simply add the lengths of the three sides together.

2. What information do I need to find the perimeter of a triangle?

To find the perimeter of a triangle, you need to know the lengths of all three sides. This can be given in the problem or you may need to measure or calculate the lengths using other information provided.

3. Can you use the Pythagorean theorem to find the perimeter of a triangle?

No, the Pythagorean theorem only applies to right triangles and only helps to find the length of one side. To find the perimeter, you need to know the lengths of all three sides.

4. What units should the perimeter of a triangle be measured in?

The perimeter of a triangle can be measured in any unit of length, such as centimeters, meters, or inches. Just make sure to keep the units consistent when adding the lengths of the sides together.

5. Can the perimeter of a triangle be negative?

No, the perimeter of a triangle cannot be negative. Perimeter is a measure of distance and distance cannot be negative. If you get a negative value when finding the perimeter, then you have likely made a mistake in your calculations.

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