1. 2.4 Isosceles Triangles, Medians, Altitudes, and Concurrent Lines
CHAPTER
2.4 Isosceles Triangles, Medians, Altitudes, and Concurrent Lines
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2. Theorem 2.5 Isosceles Base Angles Theorem
Proof
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3. Corollary 2.6 If Equilateral then Equiangular Triangle
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4. Corollary 2.8 If Equiangular then Equilateral Triangle
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Definitions
When three or more lines intersect at one point, they are concurrent. The point at which they intersect is called the point of concurrency.
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Definitions
We say the circle is circumscribed about the triangle. P, the point of concurrency of the perpendicular bisectors of this triangle, is also called the circumcenter of the triangle.
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Definitions
The circumcenter of a triangle can be inside, on, or outside a triangle. Notice the type of triangle below and each location of P, the circumcenter. Acute Right Obtuse
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8. Using the Circumcenter of a Triangle
The circumcenter of is point R. Fill in the blanks. a. RD = ____ = ____ b. RD = ________ units Solution Since R is the circumcenter, the distance from R to each vertex of the triangle is the same. RE RF 8
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Definition
A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side. A triangle’s three medians are always concurrent. The point of intersections of the medians of a triangle is the centroid.
Median
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10. Theorem 2.10 Concurrency of Medians Theorem
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11. Finding the Length of a Median
In the diagram, AC = 10 units. Find AE. Solution Point C is the centroid of the triangle because it is the point of concurrency of the medians.
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Definition
An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside or outside the triangle, or it can be a side of the triangle.
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Hint
Take care when constructing or identifying an altitude. Remember: An altitude is a segment from a vertex of the triangle perpendicular to the opposite side (or the opposite side extended).
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14. Identifying Medians and Altitudes
For triangle PQS, identify the given segments as a median, an altitude, or neither. a. b. Solution a. is a segment that extends from vertex Q to the side opposite Q. Since T is the midpoint of segment PS.
is a median
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15. Identifying Medians and Altitudes
For triangle PQS, identify the given segments as a median, an altitude, or neither. a. b. Solution b. is a segment that extends from vertex P to the line containing segment SQ, the side opposite P.
is a median
is an altitude
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Definition
The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter of a triangle can be inside, on, or outside the triangle. Acute Right Obtuse
Each blue point is the orthocenter
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17. Finding the Orthocenter of a Triangle
Page 102 Student Activity
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Definition
An angle bisector of a triangle is a bisector of an angle of the triangle.
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Definitions
The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle and the incenter is always inside the triangle. The incenter P is the center of the circle that is inscribed in the triangle. Notice that the circle touches each side once, at points X, Y, and Z. The radius of the circle is the (perpendicular) distance from P to each side.
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20. Theorem 2.12 Concurrency of Angle Bisectors Theorem
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21. Identifying and Using the Incenter of a Triangle
GE = 2x − 7 and GF = x + 4. What is GH?
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22. Identifying and Using the Incenter of a Triangle
GE = 2x − 7 and GF = x + 4. What is GH? Solution G is the incenter of the triangle because it is the point of concurrency of the angle bisectors. Using the Concurrency of Angle Bisectors Theorem, the distances from the incenter to the three sides of the triangle are equal, so GE = GF = GH. Use this relationship to find x.
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23. Identifying and Using the Incenter of a Triangle
GE = 2x − 7 and GF = x + 4. What is GH? Solution 2x – 7 = x + 4 2x = x + 11 x = 11 Now find GF. GF = x + 4 = = 15 Since GE = GF = GH, then GH = 15 units.
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Summary
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