Question

select all the statements that are true for \\( \triangle a b c \\).
a \\( \overline{a h} \\) is an altitude.
b \\( \overline{i h} \\) is a median.
c \\( \overline{j c} \\) is a median.
d. the medians and altitudes intersect at the same point.
e. the altitudes intersect outside of the triangle

Explanation:

• For option A: An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Since \(AH\perp BC\), \(\overline{AH}\) is an altitude.
• For option B: A median of a triangle is a segment from a vertex to the mid - point of the opposite side. There is no indication that \(H\) is the mid - point of any side, so \(\overline{IH}\) is not a median.
• For option C: A median is a segment from a vertex to the mid - point of the opposite side. There is no indication that \(J\) is the mid - point of \(AC\), so \(\overline{JC}\) is not a median.
• For option D: In a triangle, the medians and altitudes (in a non - degenerate triangle) intersect at the centroid (for medians) and the orthocenter (for altitudes). In an acute triangle (which \(\triangle ABC\) appears to be), the centroid and orthocenter are both inside the triangle and can be considered as intersecting at a point within the triangle.
• For option E: In an acute triangle, the altitudes intersect inside the triangle.

Answer:

A. \(\overline{AH}\) is an altitude; D. The medians and altitudes intersect at the same point.