Grade 10 · Core Mathematics · Lesson 8 of 11

Determining the Equation of a Mirror Line

A Grade 10 Core Mathematics CBE lesson from the sub-strand Reflection and Congruence. In it, learners determine the midpoint and gradient of line segments connecting me corresponding object and image points.

Strand Measurements and GeometrySub-strand Reflection and CongruenceLesson code 10CM-22-L840 minutes · 3 slides

KICD learning outcome (d)

Determine the equation of the mirror line given an object and its image.

By the end of this lesson, learners should be able to

Determine the midpoint and gradient of line segments connecting me corresponding object and image points.
Construct perpendicular bisectors to locate mirror lines on the Cartesian plane.
Calculate the equation of a mirror line in the form $y = m x + c$ given an object shape and its reflected image.

Key vocabulary

Perpendicular bisector: A line that cuts a given line segment into two equal halves at right angles.; The mirror line is the perpendicular bisector of the segment connecting point $A$ to $A^{'}$ .
Negative reciprocal: A number flipped in fraction form with its sign changed, yielding a product of $- 1$ .; The line segment gradient $- \frac{1}{2}$ gives a negative reciprocal perpendicular gradient of $2$ .
Gradient: The measure of steepness of a line calculated as vertical change divided by horizontal change.; The segment $K K^{'}$ has a gradient of $1$ , giving a perpendicular gradient of $- 1$ .
Equidistant: Situated at an equal distance from a specified reference point, line, or surface.; Point $T (1, 3)$ and $T^{'} (5, - 1)$ are equidistant from $y = x - 2$ .

Lesson preview

Slide 1 · Finding the Mirror Line between Object and Image PointsModel · 12 min

Figure 1: A coordinate plane showing object point A and image A' connected by a dashed line, with their perpendicular bisector representing the mirror line.

To determine the equation of a mirror line when an object and its reflected image are given, we locate the perpendicular bisector of the line segment joining any corresponding object-image pair. The perpendicular bisector passes through the midpoint of the segment, and its gradient is the negative reciprocal of the segment's gradient. Using the coordinate change formula $gradient = \frac{change in y}{change in x}$ , we establish the linear equation $y = m x + c$ representing the mirror line.

The mirror line acts as the perpendicular bisector of the segment connecting any object point to its corresponding image point.
The midpoint of segment $A A^{'}$ lies directly on the mirror line for every corresponding pair.
The gradient of the mirror line is the negative reciprocal of the gradient of segment $A A^{'}$ .
We substitute the midpoint coordinates into $y = m x + c$ to find the complete line equation.

Worked example. Find the midpoint

M

A A^{'}

M = (\frac{1 + 5}{2}, \frac{4 + 2}{2}) = (3, 3)

. Calculate the gradient of

A A^{'}

m_{1} = \frac{2 - 4}{5 - 1} = \frac{- 2}{4} = - \frac{1}{2}

. Find the perpendicular gradient

m_{2}

m_{2} = - \frac{1}{- 1 / 2} = 2

. Substitute

M (3, 3)

into

y - y_{1} = m_{2} (x - x_{1})

y - 3 = 2 (x - 3) ⟹ y = 2 x - 3

. Answer: $y = 2 x - 3$ .

Check question: Point $P (1, 7)$ has reflected image $P^{'} (5, 3)$ . Determine the equation of the mirror line.

$y = x + 2$
$y = - x + 8$
$y = x$
$y = x + 4$

Show answer

$y = x + 2$ . The midpoint is $M = (\frac{1 + 5}{2}, \frac{7 + 3}{2}) = (3, 5)$ . The gradient of $P P^{'}$ is $\frac{3 - 7}{5 - 1} = \frac{- 4}{4} = - 1$ . The perpendicular gradient is $1$ . Substituting into $y - y_{1} = m (x - x_{1})$ gives $y - 5 = 1 (x - 3) ⟹ y = x + 2$ .

Slide 2 · Determining Mirror Line Equations for TrianglesPractice · 14 min

Figure 2: A Cartesian plane displaying an object triangle and its reflected image, with the perpendicular bisector between corresponding vertices K and K' drawn.

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All 3 slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Determining the Equation of a Mirror Line

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Reflection and Congruence