Orthocenter and circumcentre

Geometry Level 3

Let $a,b,c$ be the length of sides of a triangle such that $a+b=17,a+c=25$ and $b+c=18$ and $l$ be the distance between circumcentre and orthocentre of the triangle, then find $8l-2$ .

The answer is 50.

1 solution

Sean Ty
Jul 29, 2014

We have the equations $\displaystyle a+b=17 \\ a+c=25 \\ b+c=18$

Solving for $a$ , $b$ , and $c$ , we get $a=12$ $b=5$ and $c=13$

By the converse of the Pythagorean Theorem, $\triangle ABC$ is right-angled at $A$ .

The orthocenter of this triangle is its vertex. Which we call $(0,0)$ . $B$ will be $(12,0)$ and $C$ will be $(0,5)$ .

Solving for the centroid of this triangle, we get that the centroid is on point $(4,\dfrac{5}{3})$

The distance between its centroid and its orthocenter is $\dfrac{13}{3}$ .

We know that the distance between the orthocenter and the centroid is twice the distance of the circumcenter and the centroid. So the distance between the orthocenter and the circumcenter is $\dfrac{13}{2}$ .

And $\displaystyle (8)(\dfrac{13}{2})-2=\boxed{50}$

Super awesome solution! :D

Jayakumar Krishnan - 6 years, 9 months ago

It was not required to find centroid as we know the location of circumcentre of a right triangle which is at the midpoint of hypotenuse.

mietantei conan - 6 years, 9 months ago

can we find without use of coordinate geometry

nikhil jaiswal - 6 years, 7 months ago

Yes...The circumcenter is the midpoint of the hypotenuse and orthocenter is point A. The Hypotenuese is length 13, so 13 0.5=6.5. The altitude to the hypotenuese is the distance between the orthocenter and circumcenter which is the square root of 6.5^2 which is 6.5. Thus, (8 6.5)-2=50 which is the final answer.

Yashas Ravi - 2 years, 1 month ago

Orthocenter and circumcentre

The answer is 50.

1 solution

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