Question

The perpendicular bisectors of sides AC and BC of ΔABC intersect side AB at points P and Q respectively, and intersect each other in the exterior (outside) of ΔABC. Find the measure of ∠ACB if m∠CPQ = 78° and m∠CQP = 62°.

Answer 1


 

The following is an interactive proof using GSP.

To proceed, click on each Step of the proof. If you want to retrace your steps you may undo your previous step by clicking on the Undo Step. 
Claim:

The three perpendicular bisectors of the sides of a triangle are concurrent.

Proof:

Step 1:



Step 2


Construct the midpoint of segment AC. Label the midpoint of AC as M.



Step 3


Construct the perpendicular bisector of segment AC. Call the perpendicular bisector p. 



Step 4


Construct any point K on the perpendicular bisector p. 



Step 5


Construct segment AK and segment KC. 



AM=MC by definition of midpoint.
KM=KM by the reflexive property.
Angle AMK and angle CMK are cut by a perpendicular line, p. Hence, the angles are right angles and are both equal to 90 degrees. So, m<AMK = m<CMK and angle AMK is congruent to angle CMK. 
Now, triangle AMK is congruent to triangle CMK by SAS. 
By CPCTC, AK=KC. 



Step 6


Construct the midpoint of segment AC. Call it M'.
Construct the perpendicular bisector of AC. Label as q. 



Since AC and BC are not parallel, the perpendicular bisectors, p and q, must also intersect.
Let point K be the point of intersection.

Step 7


Constuct point K as the point of intersection of p and q. 



Step 8


Construct segment BK. 



KC=BK using similar reasoning in Step 5. We know AK=KC. So by substitution AK=KC=BK.
Furthermore, K is taken to be the intersection of the perpendicular bisectors p and q. 
Since K is on the perpendicular bisectors pand q, it must also lie on the perpendicular bisector of AB.
Therefore, K is on the perpendicular bisector of AB.

Since K lies on the perpendicular bisectors of AC, BC, and AB, K is the point of concurrency.
Hence, we have proved that the three perpendicular bisectors of the sides of a triangle are concurrent.

The following shows the construction of the perpendicular bisector of AB, labeled as r.
The picture shows K to be the point of concurrency.