Whereas the angle bisector theorem deals with congruent angles, hence creating equal distances from the incenter to the side of the triangle. Perpendicular bisector theorem deals with congruent segments of a triangle, thus allowing for the diagonals from the vertices to the circumcenter to be congruent. What’s so confusing about these two theorems is that their drawings look almost identical, but their conclusions are different! Moreover, the three angle bisectors meet at point G, called the incenter. If h = 0, the bisectors are xy = 0 i.e.Using the triangle above, we can see that angle A is bisected by segment AF, angle B is bisected by segment BD, and angle C is bisected by segment CE, where segments AF, BD, and CE are called the angle bisectors of triangle ABC. If a = b, then bisectors are x 2 – y 2 = 0 i.e. ⇒ (x 2 –y 2) /(a–b) = xy/h is required equation of angle bisectors …… (1) Note: Let P( α, ß) be any point on one of bisectors. Let ax 2 + 2hxy + by 2 = 9 represent lines y – m 1 x = 0 and y – m 2 x = 0 We try to compute the combined equations of angle bisectors of lines represented by ax 2 + 2hxy + by 2 = 0. Hence equation of the bisector of the angle containing the point (1, 2) is 4x + 3y – 6/5 = 5x + 12y + 9/13 ⇒ 9x – 7y – 41 = 0.Ĭombined Equations of Angle Bisectors of Lines Hence, the bisector of the obtuse angle is 9x – 7y – 41 = 0 and the bisector of the acute angle is 7x + 9y – 3 = 0. If θ is the angle between the line 4x + 3y – 6 = 0 and the bisector 9x – 7y – 41 = 0, then tan θ ≥ 1. Illustration:įor the straight lines 4x + 3y – 6 = 0 and 5x + 12y + 9 = 0, find the equation of theīisector of the obtuse angle between themīisector of the acute angle between them,īisector of the angle which contains (1, 2)Įquations of bisectors of the angles between the given lines are Note:Įquation of straight lines passing through P(x 1, y 1) and which are equally inclined with the lines a 1x + b 1y + c 1 = 0 and a 2x + b 2y + c 2 = 0 are the ones which are aparllel to teh bisectors between these two lines and they pass through the point P. If |p| = |q|, then the lines L 1 and L 2 are perpendicular. If |p| |q|, then e 1 is the obtuse angle bisector. We take a point R on one of these lines and draw perpendiculars on e 1 and e 2. Suppose we have two lines L 1 = 0 and L 2 = 0 and e 1 = 0 and e 2 = 0 are the bisectors between these two lines. Īnother way of identifying the Acute Angle Bisectors and Obtuse Angle Bisectors Suppose we have two lines represented by the equationsĪ 1 x + b 1 y+ c 1 = 0 and a 2 x + b 2 y + c 2 = 0. If (x 3, y 3) ≡ (0, 0) and A 2A 1 + B 2B 1 > 0 then the bisector towards the origin is the obtuse angle bisector. Then above equation with changed equations of lines will given the required bisector. If the signs are different multiply one of the equations with ‘–1’ throughout, so that positive sign is obtained. To determine a bisector which lies in the same relative position with respect to the lines as a given point S(x 3, y 3) does, make the signs of the expressions A 1x 3 + B 1y 3+ C1 and A 2x 3 + B 2y 3 + C 2 identical. In case |tan θ| 1, then this represents the obtuse angle bisector. Compute the angle between the the initial line and one of the bisectors. It is very simple to determine whether the bisector is an acute angle bisector or an obtuse angle bisector. How can we differentiate between the Acute Angle Bisector and the Obtuse Angle Bisector? This equation gives two bisectors: one-acute angle bisector and the other obtuse bisector. Generalizing for any point (x, y), the equation of the angle bisector is obtained as: Concepts of Physics by HC Verma for JEE.IIT JEE Coaching For Foundation Classes.
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