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Core Connections Geometry, 2013
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Core Connections Geometry, 2013 View details
Chapter Closure
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Exercise 125 Page 722

a We are given that A(2,4), B(9,5), and C(4,10). Let's use these points to draw △ ABC on a graph paper.
We want to check whether the point D(3, 7) is a midpoint of AC. To do so, let's find the midpoint of AC using the Midpoint Formula. Then we will check whether its coordinates are the same as the coordinates of D.
(x_1+x_2/2,y_1+y_2/2) The coordinates of the endpoints A and C are (2, 4) and (4, 10), respectively. Let's use these to find the coordinates of the midpoint.
M(x_1+x_2/2,y_1+y_2/2)
SubstitutePoints

Substitute ( 2, 4) & ( 4, 10)

M(2+ 4/2,4+ 10/2)
AddTerms

Add terms

M(6/2,14/2)
CalcQuot

Calculate quotient

M(3, 7)
The midpoint is located at (3, 7). Since D has the same coordinates, point D is the midpoint of AC.
b An equation in slope-intercept form follows a specific format.
y= mx+b For an equation in this form, m is the slope and b is the y-intercept. Let's use the points D(3, 7) and B(9, 5) to calculate m. We will start by substituting the points into the Slope Formula.
m=y_2-y_1/x_2-x_1
SubstitutePoints

Substitute ( 3, 7) & ( 9, 5)

m=5- 7/9- 3
▼
Simplify right-hand side
SubTerms

Subtract terms

m=- 2/6
MoveNegNumToFrac

Put minus sign in front of fraction

m = - 2/6
ReduceFrac

a/b=.a /2./.b /2.

m= - 1/3
A slope of - 13 means that for every 3 horizontal step in the positive direction, we take 1 vertical step in the negative direction. Now that we know the slope, we can write a partial version of the equation. y= -1/3 x+b To complete the equation, we also need to determine the y-intercept, b. Since we know that the given points will satisfy the equation, we can substitute one of them into the equation to solve for b. Let's use ( 3, 7).
y=- 1/3x + b
SubstituteII

x= 3, y= 7

7=-1/3( 3)+b
▼
Solve for b
MoveRightFacToNumOne

1/b* a = a/b

7 = -3/3+b
CalcQuot

Calculate quotient

7=-1+b
AddEqn

LHS+1=RHS+1

8 = b
RearrangeEqn

Rearrange equation

b = 8
A y-intercept of 8 means that the line crosses the y-axis at the point (0,8). We can now complete the equation. y= - 1/3x+8
c BD is a height of △ ABC if BD is perpendicular to AC. That is the case if the product of their slopes equals -1. Let's first find the slopes of these line segments. In Part B, we found that the slope of BD is - 13. To find the slope of AC, let's recall the Slope Formula.
m = y_2 - y_1/x_2 - x_1Here, (x_1,y_1), and (x_2,y_2) are the coordinates of two point lying on a line that we calculate the slope of. We want to find the slope of AC, so we will substitute A( 2, 4) for ( x_1, y_1) and C( 4, 10) for ( x_2, y_2) in the above formula. Let's do it!
m=y_2-y_1/x_2-x_1
SubstitutePoints

Substitute ( 2,4) & ( 4,10)

m=10- 4/4- 2
SubTerms

Subtract terms

m=6/2
CalcQuot

Calculate quotient

m= 3
Finally, let's find the product of - 13, the slope of BD, and 3, the slope of AC. If the result is -1, these line segments are perpendicular. -1/3 * 3 = -1
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Exercises 125 (Page 722)
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