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Pearson Geometry Common Core, 2011

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Pearson Geometry Common Core, 2011 View details

3. Bisectors in Triangles

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Exercise 7 Page 305

To find the circumcenter, we need equations for the perpendicular bisectors of two sides of the triangle.

(- 2, -3)

Practice makes perfect

Let's start by labeling the coordinates of the vertices of the given triangle.

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 SR is horizontal, any perpendicular line will be vertical. Similarly, since RT 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
SR	( -4,0), ( 0,0)	U(-4+ 0/2,0+ 0/2)	U(-2,0)
RT	( 0,0), ( 0,-6)	V(0+ 0/2,0+( -6)/2)	V(0, -3)

Let's add these midpoints to our graph.

Given the information, we know that the perpendicular bisectors through RS and RT have the equations x=- 2 and y=- 3, 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 (- 2, -3).

Subchapter links

Lesson Check p.304

Practice and Problem-Solving Exercises p.305-307

Standardized Test Prep p.307

Mixed Review p.307