# Perpendicular Bisector of a triangle

• nmnna
In summary: The description seems confusing to me as well.Could it be that point P should be located where the perpendicular bisector of BC cuts BA?What is the answer given in your textbook?If it is close to 3.75 cm, then your last diagram is not correct regarding location of P.
nmnna
Homework Statement
##ABC## is a triangle such that ##\angle ABC = 37^{\circ}15'##, ##\angle ACB = 59^{\circ}40'##, ##BC = 8## cm; the perpendicular bisector of ##BC## cuts ##BA##, ##CA## produced at ##P, \ Q##. Find the length of ##PQ##.
Relevant Equations
##\tan(\alpha) = \frac{opposite \ side}{adjacent \ side}##
Here is my attempt to draw a diagram for this problem:

I'm confused about the "the perpendicular bisector of ##BC## cuts ##BA##, ##CA## produced at ##P, \ Q##" part of the problem.
How does perpendicular bisector of ##BC## cut the side ##CA##?

It cuts CA produced. Extend the line CA until it meets the bisector. That point is Q (not where you have put it).

nmnna and FactChecker
mjc123 said:
It cuts CA produced. Extend the line CA until it meets the bisector. That point is Q (not where you have put it).
Thank you

mjc123 said:
It cuts CA produced. Extend the line CA until it meets the bisector. That point is Q (not where you have put it).

I changed my diagram.
Now I have the right triangle ##\triangle PQC##, where ##CP = 4##cm (since ##PQ## is a perpendicular bisector), ##\angle QCP = 59^{\circ}40'##, so I can find ##PQ## using the relation $$\tan\angle QCP = \frac{PQ}{CP}$$
I got ##\approx 6.818## which is not the answer given in my textbook. Where did I go wrong?

Last edited:
nmnna said:
View attachment 280723
I changed my diagram.
Now I have the right triangle ##\triangle PQC##, where ##CP = 4##cm (since ##PQ## is a perpendicular bisector), ##\angle QCP = 59^{\circ}40'##, so I can find ##PQ## using the relation $$\tan\angle QCP = \frac{PQ}{CP}$$
I got ##\approx 6.818## which is not the answer given in my textbook. Where did I go wrong?
The description seems confusing to me as well.
Could it be that point P should be located where the perpendicular bisector of BC cuts BA?
If it is close to 3.75 cm, then your last diagram is not correct regarding location of P.

nmnna
P is the point you have called L. The original statement, which is perhaps not as clear as it might be, means "the perpendicular bisector of BC cuts BA at P and cuts CA produced at Q."

nmnna and Lnewqban
mjc123 said:
P is the point you have called L. The original statement, which is perhaps not as clear as it might be, means "the perpendicular bisector of BC cuts BA at P and cuts CA produced at Q."

## 1. What is a perpendicular bisector of a triangle?

A perpendicular bisector of a triangle is a line or segment that divides a side of a triangle into two equal parts at a 90-degree angle.

## 2. How do you construct a perpendicular bisector of a triangle?

To construct a perpendicular bisector of a triangle, you can use a compass and a straightedge. First, draw a line that passes through the midpoint of the side you want to bisect. Then, using the compass, draw arcs from both endpoints of the side that intersect with the line you just drew. The point where the arcs intersect is the center of the perpendicular bisector. Finally, draw a line through this point that is perpendicular to the side you want to bisect.

## 3. What is the significance of a perpendicular bisector in a triangle?

A perpendicular bisector has several important properties in a triangle. It passes through the midpoint of the side it bisects, divides the triangle into two congruent right triangles, and is equidistant from the endpoints of the side it bisects. Additionally, the perpendicular bisectors of the three sides of a triangle intersect at a single point called the circumcenter, which is the center of the circumcircle (a circle passing through all three vertices of the triangle).

## 4. How is the perpendicular bisector related to the concept of symmetry?

The perpendicular bisector is a line of symmetry in a triangle. This means that if you fold the triangle along the perpendicular bisector, the two halves will be identical. This also means that any point on the perpendicular bisector is equidistant from the two endpoints of the side it bisects, making it a line of reflectional symmetry.

## 5. Can a triangle have more than one perpendicular bisector?

No, a triangle can only have one perpendicular bisector for each side. This is because the perpendicular bisector must pass through the midpoint of the side it bisects, and there is only one midpoint for each side. However, the perpendicular bisectors of the three sides of a triangle can intersect at a single point, as mentioned before, creating a unique point of concurrency called the circumcenter.

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