Orthocenter Location in Acute Triangles
The orthocenter of a triangle is the point where its three altitudes intersect. An altitude is a line segment through a vertex and perpendicular to the opposite side.
An acute triangle is defined as a triangle where all three internal angles measure less than $90^\circ$.
Orthocenter Position
For an acute triangle, the intersection point of the altitudes (the orthocenter) is always located:
- Inside the triangle.
In contrast:
- The orthocenter lies outside an obtuse triangle (a triangle with one angle greater than $90^\circ$).
- The orthocenter lies on a vertex of a right-angled triangle (a triangle with one angle equal to $90^\circ$).
Therefore, for a triangle with all angles less than $90^\circ$ (an acute triangle), the orthocenter is inside the triangle.
Similar Questions
In triangle ABC, a line segment DE is drawn through the centroid G, parallel to side BC. D lies on AB and E lies on AC. What is the ratio of the area of the smaller triangle ADE to the area of the trapezoid DECB?
A circle is circumscribed about a triangle. If one of the angles of the triangle is 120°, and the radius of the circumscribed circle is r, what is the length of the side opposite the 120° angle?
In △ABC and △XYZ, AB = XY, BC = YZ, and CA = ZX. By which congruence rule are the two triangles congruent?
Two triangles are similar with sides in the ratio 3:5. What is the ratio of their areas?
In a triangle ABC, medians AD and BE intersect at G. If the length of median AD is 12 cm, what is the length of the segment AG?
Two right-angled triangular blocks, ABC and DEF, have ∠B = ∠E = 90°. If the lengths of the hypotenuses AC and DF are equal, and the sides AB and DE are equal, are the triangles congruent? If so, by what rule?
- A right-angled triangle ABC has legs AB=6 cm and BC=8 cm. An altitude BD is drawn from the vertex B to the hypotenuse AC. What is the length of the altitude BD?
- In $\triangle ABC$, a median AD is drawn to side BC. What is the ratio of the area of $\triangle ABD$ to the area of $\triangle ACD$?
The angle bisectors of triangle ABC converge at the incenter I. Given that the measure of ∠BIC is 125°, what is the measure of ∠BAC?
- For a triangle with one interior angle greater than 90°, where is its Orthocenter located?
Important Questions from Triangles, Congruence and Similarity
G is the centroid of the equilateral triangle ABC. If AB = 8√ 3 cm, then the length of AG is equal to:
If sides of a triangle are 12 cm, 15 cm and 21 cm, then what is the inradius (in cm) of the triangle?
What is the area of quadrilateral ABCD?
It is given that ΔABC ~ ΔXYZ and Area ΔABC : Area ΔXYZ = 81 : 25. If AB = 18 cm, BC = 10 cm, CA = 15 cm, then what is the side XZ (in cm)?
Sides of two similar triangles are in the ratio 4 ∶ 9. Area of these triangles are in the ratio:
