Orthocenter and Circumcenter

Geometry Level 4

We are given a triangle $ABC$ , with orthocenter and circumcenter $H$ and $O$ respectively. We know that $\angle BAC = 65^\circ$ and the distance from $O$ to $BC$ is $65$ units. Find $AH$ .

The answer is 130.

1 solution

Manuel Kahayon
Nov 21, 2016

Let $G$ be the centroid of triangle $ABC$ . We can see that $AHG$ is similar to $GOD$ , letting $D$ be the midpoint of $BC$ . Now,

$\frac{AH}{OD} = \frac{HG}{GO} = 2$ by the Euler line.

Thus, $AH = 2OD = 2(65) = \boxed{130}$

Since $OD$ is the perpendiular from $O$ to $BC$ .

AH=2Rcos (A) and distance of O from BC=Rcos (A).So answer is 2 times 65=130.

Indraneel Mukhopadhyaya - 4 years, 6 months ago

Orthocenter and Circumcenter

The answer is 130.

1 solution

0 pending reports