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Geometry: High School15 chapters | 160 lessons

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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.

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:

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.

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.

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.

And there you have your inscribed 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:

Do you see how the border of the circle touches each side of the 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.

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.

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.

And there you have your circumscribed triangle.

Let's review what we've learned.

An **inscribed triangle** is a triangle inside a circle. To draw an inscribed triangle, you first draw your triangle. Then you draw perpendicular bisectors for each side of the triangle. Where they meet is the center of your circle. Then, you take your compass and you draw a circle from the center so that the border of the circle touches each vertex of the triangle.

A **circumscribed triangle** is a triangle with a circle inside. To draw a circumscribed triangle, you first draw a triangle. Then, you draw an angle bisector for each angle. Where these angle bisectors meet is the center of your circle. Next, you draw perpendicular lines from the center of the circle to each edge of the triangle. Where these perpendicular lines meet the edges of the triangle is where the circle touches the triangle. Then you take your compass and you draw your circle from the center that touches the edges of the triangle where the perpendicular lines are.

Topics | Explanations |
---|---|

Inscribed triangle | a triangle inside a circle with all three vertices of the triangle touching the border of the circle |

3 steps | draw triangle, draw the perpendicular bisector for each side of the triangle ,and use compass to draw circle from the center of the circle so that the border touches each vertex of the triangle |

Circumscribed triangle | a triangle with a circle inside where the border of the circle touches each side of the triangle |

4 steps | draw triangle, draw angle bisectors, draw perpendicular lines from each side of the triangle to the center of the circle, and take compass to draw circle from the center so that it touches the triangle where the perpendicular lines cross each side of the triangle |

After this lesson ends, you should have the confidence to:

- Contrast inscribed and circumscribed triangles
- Describe how to create an inscribed triangle
- Explain the steps to draw a circumscribed triangle

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Geometry: High School15 chapters | 160 lessons

- Circles: Area and Circumference 8:21
- Circular Arcs and Circles: Definitions and Examples 4:36
- Central and Inscribed Angles: Definitions and Examples 6:32
- Measure of an Arc: Process & Practice 4:51
- How to Find the Measure of an Inscribed Angle 5:09
- Tangent of a Circle: Definition & Theorems 3:52
- Measurements of Angles Involving Tangents, Chords & Secants 6:59
- Measurements of Lengths Involving Tangents, Chords and Secants 5:44
- Inscribed and Circumscribed Figures: Definition & Construction 6:32
- Arc Length of a Sector: Definition and Area 6:39
- Constructing Inscribed & Circumscribed Triangles 3:53
- Go to High School Geometry: Circular Arcs and Circles

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