Points in Triangles

Front

What is the formula to find the length of a segment in a triangle when given expressions for two segments?

Back

To find the length of a segment in a triangle when given expressions for two segments, set the expressions equal to each other and solve for the variable. Then substitute back to find the length.

Front

What is the definition of the orthocenter in a triangle?

Back

The orthocenter is the point where the three altitudes of a triangle intersect.

Front

What is a median in a triangle?

Back

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

Front

How do you find the length of a segment when given the centroid and a segment connected to it?

Back

The length of the segment from the centroid to a vertex is twice the length of the segment from the centroid to the midpoint of the opposite side.

Front

What is the circumcenter of a triangle?

Back

The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect, and it is equidistant from all three vertices.

Front

What is the relationship between the segments PB and PC when point P is the circumcenter?

Back

When point P is the circumcenter, segments PB and PC are equal in length.

Front

What is the centroid of a triangle?

Back

The centroid is the point where the three medians of a triangle intersect, and it divides each median into a ratio of 2:1.

Front

If GE is the length from the centroid to the midpoint of side AC, how do you find BE?

Back

The length BE is three times the length of GE, since the centroid divides the median in a 2:1 ratio.

Front

What is the formula for the length of a median in a triangle?

Back

The length of a median can be calculated using the formula: m = 1/2 * sqrt(2a^2 + 2b^2 - c^2), where a and b are the lengths of the sides adjacent to the median, and c is the length of the opposite side.

Front

What is the significance of the altitudes in a triangle?

Back

The altitudes of a triangle are significant because they help determine the orthocenter and can be used to calculate the area of the triangle.