Exercise 8 Page 305

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 ST is horizontal, any perpendicular line will be vertical. Examining the diagram, we can also identify the midpoint of this side.

Given the information, we know that the perpendicular bisector through ST has the equation x=0. To find the other perpendicular bisector, we need to find the equation of the line perpendicular to RT, passing through the midpoint between the R and T. This will be a 4 step process.

1. Find the slope of the line RT,
2. Find the slope of the line perpendicular to RT,
3. Find the midpoint between R and T,
4. Find the equation of the perpendicular bisector.

   Finding the Slope of RT

   To find the slope of RT, let's use the Slope Formula. m = y_1 - y_2/x_1 - x_2 The (x_1, y_1) and (x_2, y_2) are the coordinates of two points lying on RT. Since we know the coordinates of R and T, we can use them to find the slope.
   m = y_1 - y_2/x_1 - x_2

   SubstitutePoints

   Substitute ( 0,4) & ( 4,0)

   m=4- 0/0- 4

   SubTerms

   Subtract terms

   m=4/-4

   MoveNegDenomToFrac

   Put minus sign in front of fraction

   m=-4/4

   QuotOne

   a/a=1

   m=-1
   The slope is -1.

   Finding the Slope of the Line Perpendicular to RT

   By The Slopes of Perpendicular Lines Theorem, the slope of any line perpendicular to RT will be the negative reciprocal of the slope of RT, that is, -1.
   m_1 m_2 = -1

   Substitute

   m_1= -1

   -1(m_2) = -1

   DivEqn

   .LHS /(-1).=.RHS /(-1).

   m_2 = -1/-1

   QuotOne

   a/a=1

   m_2 = 1
   The slope of the perpendicular line is 1. This allows us to write the following incomplete formula for the perpendicular bisector of RT. y = x + b To find the value of b we need to find the coordinates of one point from the bisector. Since the midpoint between R and T must lie on the perpendicular bisector of RT, this will be the point we will find.

   Finding the Midpoint Between R and T

   To find the midpoint between R and T we will use the Midpoint Formula. M( x_1 + x_2/2, y_1 + y_2/2 ) Let's substitute the coordinates of R and T into this formula.
   M ( x_1 + x_2/2, y_1 + y_2/2 )

   SubstitutePoints

   Substitute ( 0,4) & ( 4,0)

   M( 0+ 4/2, 4+ 0/2)

   AddTerms

   Add terms

   M(4/2,4/2)

   CalcQuot

   Calculate quotient

   M(2,2)

   Finding the Equation of the Perpendicular Bisector

   The last step to finding the equation of the perpendicular bisector is to substitute the found coordinates for x and y in the incomplete formula for the bisector to find the value of b.
   y=x+b

   SubstituteII

   x= 2, y= 2

   2= 2+b

   SubEqn

   LHS-2=RHS-2

   0=b

   RearrangeEqn

   Rearrange equation

   b=0
   Therefore, the equation of the perpendicular bisector of RT is y=x. Having found the equation of the perpendicular bisector RT, we can graph it on the same graph we used for the perpendicular bisector of ST. Note that this line should also pass through (0, 0), because b= 0 is the y-value of the y-intercept of our line.
   

   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 (0,0).