Math 7.2 Notes

Unit 7: Locus and Concurrence

Lesson 7.2: Concurrence of Lines

• Concurrent Lines

  • Definition: 3 or more lines that meet at a single point.

  • Point of Concurrency: The point where lines converge (e.g., Point P).

Self-check Activities

• Draw concurrent lines n, m, and l; Name the point of concurrence P.

  • Vocabulary Review:

    • Perpendicular Bisector: A line that bisects another line segment at a right angle.

    • Angle Bisector: A ray that divides an angle into two congruent angles.

    • Median: A segment from a vertex to the midpoint of the opposite side.

    • Altitude: A perpendicular segment from a vertex to the opposite side.

Theorems Related to Triangle Concurrency

Theorem 7.2.1: Angle Bisectors

• The three angle bisectors of a triangle are concurrent.

  • Incenter is the point of concurrency of the angle bisectors.

  • Always located inside the triangle, regardless of its type (acute, right, obtuse).

  • Incenter serves as the center of the inscribed circle.

  • Equidistant from the three sides of the triangle (radii of the inscribed circle).

Theorem 7.2.2: Perpendicular Bisectors

• The three perpendicular bisectors of a triangle are concurrent.

  • Circumcenter is the point of concurrency of the perpendicular bisectors.

  • Can be inside (acute), outside (obtuse) or on (right) the triangle.

  • Circumcenter is the center of the circumscribed circle, equidistant from the vertices.

Theorem 7.2.3: Altitudes

• The three altitudes of a triangle are concurrent.

  • Orthocenter is the point of concurrency of altitudes.

  • Location can vary: inside (acute), outside (obtuse), or on (right).

Theorem 7.2.4: Medians

• The three medians of a triangle are concurrent.

  • Centroid is the point of concurrency of the medians; it is the center of gravity.

  • Always located inside the triangle.

  • The centroid divides each median into two segments in a 2:1 ratio (from the vertex to the midpoint).

  • If M is the midpoint of side BC, then:

    • AP = 2/3 AM

    • MP = 1/3 AM

    • AP = 2 MP

Practice & Self-Checks

Self-Check #2

1. Do you need to draw all three angle bisectors to know where the incenter is located?

2. In triangle ABC, D is the incenter. Given m∠CAB = 64° and m∠ABC = 82°, find the remaining angles.

3. Given triangle with centroid P, BF = 18 units, AP = 15 units, and PD = 5 units. Determine lengths BP, PF, CD, CP, PE, AE.

Summary of Lines and Points of Concurrency

Important Characteristics

• Circumcenter: Equidistant from vertices. Center of the circumscribed circle.

• Incenter: Equidistant from sides. Center of the inscribed circle.

• Centroid: Center of gravity. Divides each median into 1/3 and 2/3 segments.

• Orthocenter: Significant for many area-related problems.