Question

1. look for relationships consider the three types of triangles: acute, obtuse, and right. what is the relationship between the type of triangle and the location of the orthocenter? does the type of triangle tell you anything about the location of the centroid? mp.7

Explanation:

• Orthocenter:
• In an acute triangle, the orthocenter lies inside the triangle. This is because the altitudes (used to define the orthocenter) intersect within the boundaries of the acute - angled triangle.
• In a right - triangle, the orthocenter is at the vertex of the right angle. The two legs of the right - triangle are altitudes, and they intersect at the right - angled vertex.
• In an obtuse triangle, the orthocenter lies outside the triangle. Since one of the angles is greater than \(90^{\circ}\), the altitudes extended will intersect outside the triangle.
• Centroid:
• The centroid is the intersection of the medians of a triangle. For any triangle (acute, obtuse, or right), the centroid always lies inside the triangle. The centroid divides each median in a ratio of \(2:1\) from the vertex, and this property is independent of whether the triangle is acute, obtuse, or right - angled.

Answer:

• Orthocenter:
• Acute triangle: orthocenter is inside the triangle.
• Right - triangle: orthocenter is at the right - angled vertex.
• Obtuse triangle: orthocenter is outside the triangle.
• Centroid: For all types of triangles (acute, obtuse, right), the centroid is inside the triangle.