Question

in each figure below, name the centroid.
5

6

in each figure below, name the orthocenter.
7

8

Explanation:

Step1: Recall the definition of centroid

The centroid of a triangle is the point of intersection of its medians (a median is a line segment joining a vertex to the mid - point of the opposite side).

Step2: Analyze figure 5

In figure 5, since the centroid is the intersection of medians, and by the property of centroid, it is \(B\).

Step3: Analyze figure 6

In figure 6, using the property that centroid divides the median in the ratio \(2:1\) from the vertex. If we consider the median (the line segment from the vertex to the mid - point of the opposite side), and by the centroid property, it is \(B\).

Step4: Recall the definition of orthocenter

The orthocenter of a triangle is the point of intersection of its altitudes (an altitude is a perpendicular line segment from a vertex to the line containing the opposite side).

Step5: Analyze figure 7

In figure 7, since the orthocenter is the intersection of altitudes, and by the property of orthocenter, it is \(A\).

Step6: Analyze figure 8

In figure 8, using the property of orthocenter (intersection of altitudes), it is \(A\).

Answer:

1. \(B\)
2. \(B\)
3. \(A\)
4. \(A\)