We hope this detailed article on the construction of perpendicular bisector helped you in your studies. On a line segment of \(9.6\.\) with the line segment.Īns: Two lines are said to be perpendicular if they intersect such that the angles formed between them are right angles. Notice that the theorem is constructed as an "if, then" statement.The above figure shows the perpendicular bisector of a given line segment with a compass and a ruler. So, did you sink or S W I M? Converse of the Perpendicular Bisector Theorem What does that give you? Two congruent sides and an included angle, which is what postulate? The SAS Postulate, of course! Therefore, line segment S W ≅ S M. Identify S I as congruent to itself (by the reflexive property).Identify W I and I M as congruent, because they are the two parts of line segment W M that were bisected by S I.They have right angles, ∠ S I W and ∠ S I M. You now have what? Two right triangles, S W I and S I M.How can you prove that S W ≅ S M? Do you know what to do? We construct a perpendicular bisector, S I.
You will either sink or swim on this one. That means sides W H and W M are congruent, because CPCTC (corresponding parts of congruent triangles are congruent).
The Side Angle Side Postulate states, "If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent." What does that look like? We hope you said Side Angle Side, because that is exactly what it is. What do we have now? We have two right triangles, W H A and W A M, sharing side W A, with all these congruences: This means, if we run a line segment from P o i n t W to P o i n t H, we can create right triangle W H A, and another line segment W M creates right triangle W A M. That line bisected H M at 90 ° because it is a given. We are given line segment H M and we have bisected it (divided it exactly in two) by a line W A. Proving the Perpendicular Bisector Theoremīehold the awesome power of the two words, "perpendicular bisector," because with only a line segment, H M, and its perpendicular bisector, W A, we can prove this theorem. You repeat the operation at the 200 m e t e r height, and the 100 m e t e r height.įor every height you choose, you will cut guy wires of identical lengths for the left and right side of your radio tower, because the tower is the perpendicular bisector of your land.
You need guy wires a whopping 583.095 m e t e r s long to run from the top of the tower to the edge of your land. You can go out 500 m e t e r s to anchor the wire's end. So putting everything together, what does the Perpendicular Bisector Theorem say? Putting the two meanings together, we get the concept of a perpendicular bisector, a line, ray or line segment that bisects an angle or line segment at a right angle.īefore you get all bothered about it being a perpendicular bisector of an angle, consider: what is the measure of a straight angle? 180 ° that means a line dividing that angle into two equal parts and forming two right angles is a perpendicular bisector of the angle. A bisector cannot bisect a line, because by definition a line is infinite. A line is perpendicular if it intersects another line and creates right angles.Ī bisector is an object (a line, a ray, or line segment) that cuts another object (an angle, a line segment) into two equal parts. Perpendicular means two line segments, rays, lines or any combination of those that meet at right angles.
Perpendicular Bisector Theorem (Proof, Converse, & Examples)Īll good learning begins with vocabulary, so we will focus on the two important words of the theorem.
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