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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

2. Bisectors of Triangles

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Exercise 7 Page 315

To find the circumcenter, find the intersection of two perpendicular bisectors of two sides of the triangle.

(5,8)

Practice makes perfect

Let's start by graphing the triangle using the given coordinates.

To find the circumcenter, we need equations for the perpendicular bisectors of at least two sides of the triangle. Recall that a bisector cuts something in half, so we want to find lines that are perpendicular to the sides at their midpoints.

Finding Perpendicular Bisectors

By the Slopes of Perpendicular Lines Theorem, we know that horizontal and vertical lines are perpendicular. Since AB is horizontal, any perpendicular line will be vertical. Similarly, since BC is vertical, any perpendicular line will be horizontal. Let's find their midpoints. To do so, we can use the Midpoint Formula.

Side	Points	M(x_1+x_2/2,y_1+y_2/2)	Midpoint
AB	( 2,6), ( 8,6)	D(2+ 8/2,6+ 6/2)	D(5,6)
BC	( 8,6), ( 8,10)	E(8+ 8/2,6+ 10/2)	E(8,8)

Let's add these midpoints to our graph.

Given the information, we know that the perpendicular bisectors through AB and BC have the equations x=5 and y=8, respectively.

Finding the Circumcenter

The triangle's circumcenter is the point at which the perpendicular bisectors intersect.

We can see that the circumcenter is located at (5,8).

Subchapter links

Communicate Your Answer p.309

Monitoring Progress p.311-314

Exercises p.315-318