SME
Grade 10 Math · Problem type
The circle through the corners of a right triangle
Use this for “A right triangle has legs 5 and 12. Find the radius of the circle through its corners.”
Diagram
Each perpendicular bisector crosses its side at the middle, at right angles.
They meet at the circumcenter O, the center of the circle through A, B and C.
This is an example. Type your own values to start a new problem.
c
=
a
2
+
b
2
R
=
c
2
Formulas
c
=
√
a
2
+
b
2
13
=
√
5
2
+
12
2
R
=
c
2
6.5
=
13
2
Assumptions
•
The perpendicular bisectors of the sides meet at the circumcenter O, the same distance R from all three corners.
•
In a right triangle O is the midpoint of the hypotenuse, so R = c ÷ 2.
•
For other triangles, construct the bisectors: O is where they meet.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
a = 5, b = 12
FIND
hypotenuse AB (c), circumradius R (R)
Step 1 · Find hypotenuse AB (c)
Use c = √(a² + b²)
The Pythagorean theorem gives the hypotenuse.
c
=
√
5
2
+
12
2
c
=
√
25 + 144
c
=
√
169
c
=
13
Step 2 · Find circumradius R (R)
Use R = c ÷ 2
The circumcenter is the midpoint of the hypotenuse.
R
=
13
2
R
=
6.5
Check
Put the numbers back in. Both sides should match.
13
=
√
5
2
+
12
2
✓
6.5
=
13
2
✓
Topic check
See what you know: 5 questions on this topic
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