Question

Triangle ABC is the image of SABC under a reflection. Given

A(-2,5),
B(0.9).
C(3,7),
AY5,
2). B19,0), and C47,3),
what is the line of reflection?

Answer 1

The line of reflection for triangle ABC, where A(-2,5), B(0,9), C(3,7), A'(5,2), B'(9,0), and C'(7,3), is the perpendicular bisector of the segment joining each point and its image. The equation of the line of reflection is (y = -x + 6).

Step-by-step explanation:

In order to determine the line of reflection, we need to find the equation of the perpendicular bisector of the segment joining each point and its image. The midpoint of each corresponding pair of points (A and A', B and B', C and C') gives a point on the line of reflection. Let's take points A and A' as an example.

The midpoint M of the segment joining A and A' is given by the coordinates:

`\[ M \left(((-2 + 5))/(2), ((5 + 2))/(2)\right) = (1.5, 3.5) \]`

The slope of the line containing A and A' is given by:

`\[ m = ((2 - 5))/((5 - (-2))) = -(1)/(3) \]`

The negative reciprocal of this slope gives us the slope of the perpendicular bisector:

`\[ m_{\text{perpendicular}} = 3 \]`

Now, we use the slope-point form of a line to find the equation of the perpendicular bisector, substituting the coordinates of the midpoint (1.5, 3.5):

y - 3.5 = 3(x - 1.5)

Simplifying, we get the equation of the line of reflection:

y = -x + 6

Repeat this process for points B and B', as well as C and C', and you'll find the same equation for each, confirming that the line is indeed the reflection axis.