Given: bisects and TEST 534

In an isosceles triangle, the two sides of equal length are legs, and the third side is the base. See Figure 3.22. The point at which the two legs meet is the vertex of the trian- gle, and the angle formed by the legs and opposite the base is the vertex angle. The two remaining angles are base angles. If in Figure 3.23, then is isosceles with legs and , base , vertex C, vertex angle C, and base angles at A and B. With , we see that the base of this isosceles triangle is not neces- sarily the “bottom” side. AB AC ⬵ BC AB BC AC 䉭ABC AC ⬵ BC Isosceles Triangle Vertex, Legs, and Base of an Isosceles Triangle Base Angles Vertex Angle Angle Bisector Median Altitude Perpendicular Bisector Auxiliary Line Determined, Overdetermined, Undetermined Equilateral and Equiangular Triangles Perimeter Isosceles Triangles 3.3 KEY CONCEPTS Leg Leg Vertex Vertex Angle Base Base Angles Figure 3.22 A a 1 2 D B ⬔ 1 ⫽ ⬔2, so AD is the angle-bisector of ⬔BAC in ⌬ABC ~ C Figure 3.23 M is the midpoint of BC, so AM is the median from A to BC B C A M b AE BC, so AE is the altitude of ⌬ABC from vertex A to BC A E B C c M is the midpoint of BC and FM BC, so FM is the perpendicular bisector of side BC in ⌬ABC M F B C A d Consider in Figure 3.23 once again. Each angle of a triangle has a unique angle bisector, and this may be indicated by a ray or segment from the vertex of the bisected angle. Just as an angle bisector begins at the vertex of an angle, the median also joins a ver- tex to the midpoint of the opposite side. Generally, the median from a vertex of a triangle is not the same as the angle bisector from that vertex. An altitude is a line segment drawn from a vertex to the opposite side such that it is perpendicular to the opposite side. Finally, the perpendicular bisector of a side of a triangle is shown as a line in Figure 3.23d. A seg- ment or ray could also perpendicularly bisect a side of the triangle. In Figure 3.24, is the bisector of ; is the altitude from A to ; M is the midpoint of ; is the median from A to and is the perpendicular bisector of . An altitude can actually lie in the exterior of a triangle. In Figure 3.25 on page 146, which shows obtuse triangle , the altitude from R must be drawn to an exten- sion of side . Later we will use the length h of the altitude and the length b of side in the following formula for the area of a triangle: Any angle bisector and any median necessarily lie in the interior of the triangle. A = 1 2 bh ST RH ST 䉭RST BC Í FM BC ; AM BC BC AE ∠BAC AD 䉭ABC B E C D M F A Figure 3.24