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### Course: Class 9 (Old)>Unit 8

Lesson 5: Inscribed shapes problem solving

# Inscribed shapes: find diameter

Find the diameter of a circle using an inscribed right triangle. Created by Sal Khan.

## Want to join the conversation?

• Once you found out that the triangle is a right triangle, couldn't you use the Pythagorean triplet 8, 15, 17 to find the hypotenuse and diameter?
• That's exactly what Sal did. He determined it was a right triangle, then did 8^2 + 15^2 = x^2 and got 17.
• How did Sal know that angle C was a right angle?
• There is a theorem in geometry that says for any triangle with one side completely on the diameter of its circumscribed circle (the circle touching all three vertices of the triangle), then this triangle must be a right triangle, with the right angle where the two shorter lines of the triangle meets the circle.
It would be a good exercise for you to prove it, if you haven't already, as it's fairly easy to prove.
• Is the hypothenuse of an inscribed right triangle always a diameter?
• Yes, always. "If a right triangle is inscribed in a circle, then its hypotenuse is a diameter of the circle. If one side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle."

That quote is from here:
http://www.math.uakron.edu/amc/Geometry/HSGeometryLessons/InscribedRightTriangles.pdf
• Did anyone else do a completely different method to find that C was a right angle? I sort of forgot this was in the inscribed and circumscribed angle series, so I didn't do it that way.
• Which method did you use? I've been trying to come up with another one because of your comment but can't seem to figure it out :D.
• So how do we find angles when there are no side lengths?
• The only way to identify angles without side lengths that I know of is to use a protractor or means of indirect measurement, such as subtracting the two other angles from 180 degrees.
• At , how can angle ACB is a right angle if we don't know if that angle is exactly 90 degrees? Like that could be an 89.7-degree angle.
• You know that if you have a diameter, it separates the circle into two 180 degree arcs. An inscribed angle is always 1/2 of the arc that it subtends. So 1/2*180=90.
• i still do not get how < C is 90.care to explain?
• The central angle and the inscribed angle both subtend the same arc. since the central angle is 180 deg and the inscribed angle in 1/2 of than the measure of angle c was 90 deg.
• How did sal get 17? I understand the rest but 17 appeared and sal didn’t explain how to get 17.