"Perpendicular bisectors of line segments" PPT courseware 3

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"Perpendicular bisectors of line segments" PPT courseware 3

Properties of perpendicular bisectors of line segments

A point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the line segment.

What is the flaw in the above proof when point P coincides with point C?

△PCA and △PCB will not exist.

Are PA and PB still equal?

equal!

At this time, PA=CA, PB=CB

It is known that AC=CB ∴PA=PB

Known: line segment AB, and PA=PB

Prove: Point P is on the perpendicular bisector MN of line segment AB.

Prove: Let PC⊥AB pass through the point P and the perpendicular foot is C.

In Rt△PCA and Rt△PCB

PA=PB, PC=PC

∴ △PCA ≌ △PCB(HL)

∴AC=BC

∴PC is the perpendicular bisector of line segment AB. That is, point P is on the perpendicular bisector MN of line segment AB.

Converse theorem

A point equidistant from the two endpoints of a line segment is on the perpendicular bisector of the line segment.

summary:

1. A point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the line segment.

2. A point equidistant from the two endpoints of a line segment is on the perpendicular bisector of the line segment.

The perpendicular bisector of a line segment can be viewed as

The set of all points that are equidistant from the two endpoints of a line segment.

Proof question: 1. Known: In ABC, ∠C=90°, ∠A=30°, BD

Divide △ABC equally into AC and D.

Prove: Point D is on the perpendicular bisector of AB.

Prove: ∵ ∠C=90°, ∠A=30° (known)

∴ △ABC=60° (Sum of interior angles of a triangle theorem)

∵BD bisects ∠A BC (known)

∴ ∠ABD=30o (definition of angle bisector)

∴ ∠A=∠ABD (equivalent substitution)

∴ AD=BD (equiangular opposite sides)

∴ Point D is on the perpendicular bisector of AB. (A point that is equidistant from the two endpoints of a line segment is on the perpendicular bisector of the line segment.)

summary:

A point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the line segment.

A point equidistant from the two endpoints of a line segment is on the perpendicular bisector of the line segment.

The perpendicular bisector of a line segment can be viewed as the set of all points that are equidistant from the two endpoints of the line segment.

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"Perpendicular bisectors of line segments" PPT courseware 10:

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"Perpendicular bisectors of line segments" PPT courseware 9:

"Perpendicular Bisector of a Line Segment" PPT Courseware 9 Learning Objectives 1. Be able to use the ruler and compass method to draw a perpendicular bisector of a known line segment, and be able to prove its correctness. 2. Experience the process of exploration, prove the property theorem of perpendicular bisectors of line segments and its inverse theorem, and further develop...

"Perpendicular bisectors of line segments" PPT courseware 8:

"Perpendicular Bisectors of Line Segments" PPT Courseware 8 Teaching Objectives 1. Be able to tell the theorem and converse theorem of perpendicular bisectors of line segments, and be able to use these two theorems differently. 2. Understand the application of mathematics learning methods, observation, generalization, verification, comparison, etc. in this lesson. 3..

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Update Time: 2024-09-30

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