5.3. Angle Bisectors in Triangles http://www.ck12.org
TABLE5.4:
Statement Reason
1.- The shortest distance from a point to a line is perpen-
dicular.
3.^6 DABand^6 DCBare right angles
4.^6 DAB∼=^6 DCB
5.BD∼=BD - 4 ABD∼= 4 CBD
- CPCTC
−→
BDbisects^6 ABCDetermine if the following descriptions refer to the incenter or circumcenter of the triangle.
- A lighthouse on a triangular island is equidistant to the three coastlines.
- A hospital is equidistant to three cities.
- A circular walking path passes through three historical landmarks.
- A circular walking path connects three other straight paths.
Constructions
- Construct an equilateral triangle.
- Construct the angle bisectors of two of the angles to locate the incenter.
- Construct the perpendicular bisectors of two sides to locate the circumcenter.
- What do you notice? Use these points to construct an inscribed circle inside the triangle and a circumscribed
circle about the triangle.
Multi- Step Problem
- Draw^6 ABCthroughA( 1 , 3 ),B( 3 ,− 1 )andC( 7 , 1 ).
- Use slopes to show that^6 ABCis a right angle.
- Use the distance formula to findABandBC.
- Construct a line perpendicular toABthroughA.
- Construct a line perpendicular toBCthroughC.
- These lines intersect in the interior of^6 ABC. Label this pointDand draw
−→
BD.
- Is
−→
BDthe angle bisector of^6 ABC? Justify your answer.