Question

practice with orthocenters and centroids. which figure has an orthocenter outside the triangle?

Explanation:

Step1: Recall orthocenter rules

The orthocenter of a triangle lies outside the triangle if and only if the triangle is obtuse (has one angle greater than \(90^\circ\)). For acute triangles (all angles \(< 90^\circ\)) and right triangles (one angle \(= 90^\circ\)), the orthocenter is inside (acute) or at the right - angled vertex (right).

Step2: Analyze each triangle

• First triangle: Angles are \(30^\circ\), \(75^\circ\), \(75^\circ\). All angles \(< 90^\circ\) (acute triangle), orthocenter inside.
• Second triangle: Angles \(54^\circ\), \(56^\circ\), \(74^\circ\). All angles \(< 90^\circ\) (acute triangle), orthocenter inside.
• Third triangle: Angles \(45^\circ\), \(110^\circ\), \(25^\circ\) (since \(45 + 110+25=180\)). The angle \(110^\circ>90^\circ\) (obtuse triangle), so orthocenter outside.
• Fourth triangle: Has a right angle (\(90^\circ\)) (right triangle), orthocenter at the right - angled vertex.

Answer:

The triangle with angles \(45^\circ\), \(110^\circ\), and \(25^\circ\) (the third triangle in the given figures) has an orthocenter outside the triangle.