Exercise 44 Page 560

We want to prove that the point (b,a) is the image of (a,b) after a reflection on the line y=x. To do so, we need to prove two things.

1. The distance to the line of reflection is the same for both (a,b) and (b,a).
2. The segment that connects (a,b) and (b,a) is perpendicular to the line of reflection.

Let's prove these two things one at a time.

Distance to y=x

Proving that the two points have the same distance to the line of reflection, is the same as proving that their midpoint lies on the line. Let's find the midpoint between (a,b) and (b,a) using the Midpoint Formula.

M(x_1+x_2/2,y_1+y_2/2)

SubstitutePoints

Substitute ( a,b) & ( b,a)

M(a+ b/2,b+ a/2)

CommutativePropAdd

Commutative Property of Addition

M(a+b/2,a+b/2)

Both x- and y-coordinates are the same, which means that the midpoint lies on y=x.

Perpendicularity

The line of reflection is written in slope-intercept form. y=x ⇔ y= 1x+0 We see that the slope of this line is 1. Let's now use the Slope Formula to find the slope of the segment that connects (a,b) and (b,a).

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( a,b) & ( b,a)

m=a- b/b- a

▼

Simplify right-hand side

FactorOut

Factor out -1

m=-1(- a+b)/b-a

CommutativePropAdd

Commutative Property of Addition

m=-1(b-a)/b-a

CancelCommonFac

Cancel out common factors

m=-1

The slope of this segment is - 1. Recall that the slopes of perpendicular lines are opposite reciprocals. Since 1 and - 1 are opposite reciprocals, the line y=x and the segment whose endpoints are (a,b) and (b,a) are perpendicular.

Conclusion

The distance from (a,b) to the line y=x is the same as the distance from (b,a) to the same line. The segment that connects these two points is perpendicular to the line of reflection. Therefore, (b,a) is the image of (a,b) after a reflection on the line y=x.