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Constructing Inscribed & Circumscribed Triangles

Constructing Inscribed & Circumscribed Triangles
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  • 0:01 An Inscribed Triangle
  • 0:34 Constructing An…
  • 1:19 A Circumscribed Triangle
  • 1:37 Constructing a…
  • 2:44 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

View this lesson to learn how you can construct inscribed and circumscribed triangles. Learn how both involve circles. All you need are a ruler and a compass.

An Inscribed Triangle

In this lesson, we'll see how we can draw two different circles by using a triangle. The first type is called an inscribed triangle. This type of triangle is a triangle inside a circle. All three vertices of the triangle touch the border of the circle. It looks like this:

inscribed triangle

Can you see that the three vertices of the triangle all touch the border of the circle? Whenever you see a triangle being surrounded by a circle like this, you are looking at an inscribed triangle.

Constructing an Inscribed Triangle

To draw or construct an inscribed triangle yourself, you'll need a ruler and a compass. There are three steps you need to follow. In the first step, you draw your triangle.

In the second step, you draw the perpendicular bisector for each side of the triangle. The perpendicular bisector is the line that is perpendicular to the side of the triangle and that cuts the side in half. You will see that your three perpendicular bisectors intersect at just one point. This is the center of your circle.

inscribed triangle

In the third step, you take your compass and you draw your circle from the center of the circle so that the border touches each vertex of the triangle.

inscribed triangle

And there you have your inscribed triangle.

A Circumscribed Triangle

The second type is called a circumscribed triangle. This type is a triangle with a circle inside. The border of the circle touches each side of the triangle. A circumscribed triangle looks something like this:

circumscribed triangle

Do you see how the border of the circle touches each side of the triangle?

Constructing a Circumscribed Triangle

To construct a circumscribed triangle, you need the same tools that you used to make your inscribed triangle. You'll need a ruler and a compass just like before. To draw this type of circle that gives you a circumscribed triangle, you'll need to follow four steps. In the first step, just like before, you draw your triangle.

In the second step, instead of drawing your perpendicular bisectors, you'll now draw your angle bisectors. Your angle bisector is the line that cuts your angle in half. You'll draw one for each angle. Your three angle bisectors will intersect at one point. This point is the center of your circle.

circumscribed triangle

In the third step, you draw perpendicular lines from each side of the triangle to the center of the circle. Where these perpendicular lines meet the sides of the triangle is where the circle touches the triangle.

circumscribed triangle

In the fourth step, you take your compass and you draw your circle from the center so that it touches the triangle where the perpendicular lines cross each side of the triangle.

circumscribed triangle

And there you have your circumscribed triangle.

Lesson Summary

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