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Points of Concurrency
Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency.
Example:
M is the point of concurrency of lines M w, y, and x. w x y The Four Centers of a Triangle In a triangle, the following sets of lines are concurrent: · The three medians. · The three altitudes. · The perpendicular bisectors of each of the three sides of a triangle. · The three angle bisectors of each angle in the triangle. Concurrency of the Medians The median of a triangle is the line segment that joins the vertex to the midpoint of the opposite side of the triangle. The three medians of a triangle are concurrent in a point that is called the centroid. There is a special relationship that involves the line segments when all of the three medians meet. The distance from each vertex to the centroid is two-thirds of the length of the entire median drawn from that vertex Let's Take a Look at the Diagram.... O is the centroid of Δ ABC, points D, F, and E are midpoints.
In addition, the distance from each centroid to the opposite side (midpoint) is one-third of the distance of the entire median. O is the centroid of Δ ABC, points D, F, and E are midpoints. The centroid also divides the median into two segments in the ratio 2:1, such that: and (^) and If you notice, the bigger part of the ratio is the segment that is drawn from the vertex to the centroid. The smaller part of the median is always the part that is drawn from the centroid to the midpoint of the opposite side. When working with these ratios, it is important to never mix the two up!!!
- In ΔRST, medians TM and SP are concurrent at point Q. If TQ = 3x-1 and QM = x+1, what is the length of median TM?
- In ΔABC, points J, K, and L are the midpoints of sides AB, BC, and AC, respectively. If the three medians of the triangle intersect at point P and the length of LP is 6, what is the length of BL?
- In triangle ABC, medians AD, BE, and CF are concurrent at point P. If AD = 24 inches, find the length of AP.
Examples:
- In the diagram Jose found centroid P by constructing the three medians. He measured CF and found it to be 6 inches. If PF = x, then what equation can be used to find the value of x? (1) x + x = 6 (2) 2x + x = 6 (3) 3x + 2x = 6 (4) x + (2/3)x = 6
More Examples!!!
Properties of the Circumcenter The circumcenter is the center of the circle that can be circumscribed around the triangle.
Examples
- The perpendicular bisectors of ΔABC intersect at point P. If AP = 20 and BP = 2x+4, then what is the value of x?
- The perpendicular bisectors of ΔABC intersect at point P. AP = 5 + x, BP = 10, and CP = 2y. Find x and y.
- The perpendicular bisectors of ΔABC are concurrent at P. AP = 2x - 4, BP = y + 6, and CP = 12. Find x and y. Concurrency of the Angle Bisectors An angle bisector is a line segment with one endpoint on any vertex of a triangle that extends to the opposite side of the triangle and bisects the angle. There are three angle bisectors of a triangle. The three angle bisectors of a triangle are concurrent in a point equidistant from the sides of a triangle. The point of concurrency of the angle bisectors of a triangle is known as the incenter of a triangle. The incenter will always be located inside the triangle.
Properties of the Incenter
The incenter is the center of a circle that is inscribed inside a triangle.
Concurrency of the Altitudes An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side. There are three altitudes in a triangle. The altitudes of a triangle, extended if necessary, are concurrent in a point called the orthocenter of the triangle. Location of the Orthocenter The orthocenter can fall in the interior of the triangle, on the side of the triangle, or in the exterior of the triangle. Right Triangle The orthocenter is located on the right angle. Acute Triangle The orthocenter is located inside the triangle.
Obtuse Triangle
The orthocenter is located outside the triangle.
