MHB Distance Between Intersection Points of Bisectors & Medians in Right Triangle

[Elena1]

The legs of chateti of a right triangle are 9 and 12 cm. Find the distance between the intersection point of bisectors and the point of intersection of the medians

 

[Evgeny.Makarov]

The legs of chateti of a right triangle

It should say simply "legs", or "catheti". See Wikipedia for terminology.

Find the distance between the intersection point of bisectors and the point of intersection of the medians

Do you mean angle bisectors or perpendicular bisectors?

Also, what are your thoughts about the solution? What topic does this problem belong to, e.g., coordinate geometry, circumscribed circle, etc.?

 

[Elena1]

I mean the distance between the intersection point of bisectors and the intersection point of medians

 

[Evgeny.Makarov]

I mean the distance between the intersection point of bisectors and the intersection point of medians

Do you mean angle bisectors or perpendicular bisectors?

Also, what are your thoughts about the solution? What topic does this problem belong to, e.g., coordinate geometry, circumscribed circle, etc.?

...

 

[Elena1]

...

я не знаю

 

[Evgeny.Makarov]

Вы имеете в виду биссектрисы? По-английски они называются angle bisectors, в то время как серединные перпендикуляры называются perpendicular bisectors.

If you mean angle bisectors, then Wikipedia says the radius of the inscribed circle is $r=(a+b-c)/2$ where $a$ and $b$ are legs and $c$ is the hypotenuse. Thus, if we arrange the triangle so that its right angle is in the origin and legs go along the $x$ and $y$ axes, respectively, then the coordinates if the inscribed circle center (which is also the intersection point of angle bisectors) will be $(r,r)$. The hypotenuse ends will have coordinates $(12,0)$ and $(0,9)$, so it is possible to find the middle $D$ of that segment. The intersection point of medians lies $2/3$ of the way from $(0,0)$ to $D$. This way you can find the coordinates of the intersection of medians. Then find the distance according to the usual formula for two points with known coordinates.

 

[Elena1]

Вы имеете в виду биссектрисы? По-английски они называются angle bisectors, в то время как серединные перпендикуляры называются perpendicular bisectors.

If you mean angle bisectors, then Wikipedia says the radius of the inscribed circle is $r=(a+b-c)/2$ where $a$ and $b$ are legs and $c$ is the hypotenuse. Thus, if we arrange the triangle so that its right angle is in the origin and legs go along the $x$ and $y$ axes, respectively, then the coordinates if the inscribed circle center (which is also the intersection point of angle bisectors) will be $(r,r)$. The hypotenuse ends will have coordinates $(12,0)$ and $(0,9)$, so it is possible to find the middle $D$ of that segment. The intersection point of medians lies $2/3$ of the way from $(0,0)$ to $D$. This way you can find the coordinates of the intersection of medians. Then find the distance according to the usual formula for two points with known coordinates.

could you solve my problem please i don`t understand the drawing

 

[Evgeny.Makarov]

Here is a picture that uses notations from post #7.

https://www.physicsforums.com/attachments/3563._xfImport

could you solve my problem please

No, according to rule 11 http://mathhelpboards.com/rules/ you are supposed to show some effort.