Math  /  Geometry

Questiona true
5 If line pp is a perpendicular bisector for ABC\triangle A B C, what are two conclusions you can draw.

Studdy Solution

STEP 1

What is this asking? We need to find two facts that are true when a line *p* perpendicularly bisects a side of triangle *ABC*. Watch out! Don't confuse a perpendicular bisector with an angle bisector or a median.
A perpendicular bisector must do *both* jobs: be perpendicular *and* bisect (cut in half) a side.

STEP 2

1. Define perpendicular bisector
2. First conclusion
3. Second conclusion

STEP 3

Let's imagine our triangle *ABC* and a line *p* that's a perpendicular bisector of one of its sides.
Let's say it's side *AB*.

STEP 4

Because *p* is a **perpendicular** bisector, it intersects *AB* at a **right angle** (9090^\circ).

STEP 5

And because *p* is a perpendicular **bisector**, it cuts *AB* into **two equal segments**.
So, if *M* is the point where *p* intersects *AB*, we know AM=MBAM = MB.

STEP 6

Any point on the perpendicular bisector of a segment is **equidistant** (the same distance) from the endpoints of the segment.

STEP 7

In our case, any point on line *p* is the same distance from *A* and *B*.
So, we can say that any point on line *p* is equidistant from points *A* and *B*.

STEP 8

If the perpendicular bisector *p* of side *AB* passes through the opposite vertex *C*, then triangle *ABC* must be **isosceles**.

STEP 9

This is because *C* would be a point on the perpendicular bisector, and as we just saw, any point on the perpendicular bisector is equidistant from the endpoints of the segment it bisects.

STEP 10

So, if *p* goes through *C*, then AC=BCAC = BC, which is the definition of an **isosceles triangle** (two sides are equal).

STEP 11

Two conclusions we can draw if line *p* is a perpendicular bisector of a side of ABC\triangle ABC are:
1. Any point on line *p* is equidistant from the endpoints of the side it bisects.
2. If *p* passes through the opposite vertex, the triangle is isosceles.

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