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### Course: Geometry (all content)>Unit 4

Lesson 4: Perpendicular bisectors

# Circumcenter of a right triangle

Showing that the midpoint of the hypotenuse is the circumcenter. Created by Sal Khan.

## Want to join the conversation?

• how do you find the circumradius of a right triangle
• The smart-aleck answer: find the circumcenter of the triangle, then calculate the radius to one of the vertices.

The deeper answer: the circumcenter of a right triangle is the midpoint of the hypoteneuse. Thus the circumradius is half the length of the hypoteneuse.
• what does ratio mean in this video?
• The ratio used in this video was 1/2. I believe he use AB and BM.
1/2 over
BM/AB
In other words, if AB is equal to 2, then BM is equal to 1. If BM is equal to 5, AB is equal to 10.
• At what point did he prove OC was equal to OA and OB?
• Well, he proved this just about at
Here is the logical argument: He has just proved that O is the midpoint of the hypotenuse, which means that OA = OB.
Then,
he reminds us of the initial construction of a perpendicular bisector to the leg BC.
A point on a perpendicular bisector is `equidistant` from the endpoints of the bisected segment. (I just watched a video where he proved that.) Therefore, the distance from O to B will equal the distance from O to C.

In other words, that means that line segment OB = OC
Therefore `OA = OB = OC`

This is a reconstruction of the proof that OB would be congruent to OC:
Consider a line segment, PQ
construct a perpendicular bisector of PQ with line segment RS passing through point M, the midpoint of PQ.
Connect P and Q with R to make two back-to-back triangles. Now the original line segment PQ has been divided into two equal segments PM and MQ. These are congruent. A perpendicular bisector forms an angle of 90 degrees with the original line segment, so <RMQ = <RMP. The triangles we have constructed RMQ and RMP are congruent by SAS congruency postulate.
Finally, RQ must be congruent with RP by CPCTC
This is the proof that a point on the perpendicular bisector of a line segment would be equidistant from the points at the ends of the bisected line segment.
• what is cross multiplaction?
• Hi Deepti,
Cross multiplication is one of the ways to solve an equation when it is in the form of "a fraction equals a fraction". For example:
5/x = 22/3
We are trying to isolate x so one of the ways is to use our standard algebra methods. We have fractions so we want to get rid of those first. We'll get rid of the 3 by multiplying each side by 3:
(5/x)(3/1)=(22/3)(3/1)
15/x = 22
Now lets get rid of the x in the denominator:
(15/x)(x/1)=(22/1)(x/1)
15 = 22x
Finally, we divide by 22 to isolate the x:
15/22 = 22x/22
15/22 = x
The shortcut to all of this is to cross multiply. When we cross multiply, we multiply the numerator of one side with the denominator of the other side and vice versa:
5/x = 22/3
(5)(3) = (22)(x)
15 = 22x
This looks familiar :-)
So,
15/22 = x
Hope this help[s you to understand how cross multiplication works.
Good luck!
• Doesn't a bisector have to start from the center an angle? Sal drew it in the middle of the right triangle, not from the vertex from an angle. With the positions of angles in a right triangle, the only possible perpendicular bisector is the one protruding from the right angle.
• You are talking about the angle bisector.
• Couldn't you just draw it out? Wouldn't that be easier?
• It would, but drawings aren't exact, and you wouldn't know for sure that it bisects the hypotenuse
(1 vote)
• How does he get the ratio and use it?
(1 vote)
• OM is parallel to AC. Since both segments are perpendicular to BC, each triangle is a right triangle. Since they share an angle (B), and each have a 90 degree angle, the remainin angles, A and MOB must be congruent. Two triangles are similar if their corresponding angles are congruent. The corresponding sides of similar triangles are proportional.
• What does Circumcenter mean?
(1 vote)
• The circumcenter is a two-part definition. Let's break it down, circum- means "circle". And center means "well a point that falls in the middle". So circumcenter is more like, the center of a circle. I know we are working on a triangle, but if you drew many circumradii from the circumcenter of the triangle, you are forming a circle on which the the triangle's vertices are touching.