"Perpendicular bisectors of line segments" PPT courseware 6

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"Perpendicular bisectors of line segments" PPT courseware 6

Reflection

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

Can you prove this conclusion?

Known: As shown in the figure, AC=BC, MN⊥AB is at point C, and P is any point on MN.

Prove: PA=PB.

Proof: Because MN⊥AB (known)

So ∠PCA=PCB=90° (vertical definition)

In �PCA and �PCB, because

AC=BC , (known)

∠PCA=PCB, (certified)

PC=PC, (common edge)

So △PCA≌△PCB(SAS)

Therefore PA=PB (corresponding sides of congruent triangles are equal).

Basic pass

1. As shown in the figure, it is known that AB is the perpendicular bisector of line segment CD, and E is a point on AB. If EC=7cm, then ED=______cm; if ∠ECD=600, then ∠EDC=______0.

2. As shown in the figure, in △ABC, it is known that AC=27, the perpendicular bisector of AB intersects AB at point D, and intersects AC at point E. The perimeter of △BCE is equal to 50. Find the length of BC.

Conversely: Is a point equidistant from both endpoints of a line segment on the perpendicular bisector of the line segment?

Theorem: A point equidistant from both endpoints of a line segment lies on the perpendicular bisector of the line segment.

As shown in the figure, ∵PA=PB (known),

∴Point P is on the perpendicular bisector of AB

(The point that is equidistant from the two endpoints of a line segment is at the perpendicular to the line segment.

on the straight bisector).

Theorem: The distance between a point on the perpendicular bisector of a line segment and the two endpoints of the line segment is equal.

Theorem: A point equidistant from the two endpoints of a line segment lies on the perpendicular bisector of the line segment.

Trial test

As shown in the figure, in the right angle �ABC, ∠C=90°, AD bisects ∠BAC and intersects BC at D, and DE bisects the hypotenuse AB perpendicularly, then

(1)DE=CD? Why?

(2)AD=BD? Why?

As shown in the figure, on �ABC, the known point D is on BC, and BD+AD=BC.

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

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Update Time: 2024-11-14

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