Skip to main content

Grade 10 Mathematics: Learners’ Book

INNODEMS

Section 2.2 Reflection and Congruence

Subsection 2.2.1 Identifying Lines of Symmetry Given an Object

Work in groups

Activity 2.2.1.

From a piece of paper, cut out a square shape and label it as shown below.
Square
Fold the square in half from left to right such that the corner \(A\) aligns with corner \(B\) and corner \(D\) aligns with corner \(C.\) This will create a rectangle.
Unfold the paper and make a dotted line across the fold line and label it as \(XY\) as shown.
Square
Notice that the left side of the line \(XY\) and the right side are exacly the same/ identical.
Now, fold the square in half from the top to the bottom such that the corner \(D\) aligns with corner \(A\) and corner \(C\) aligns with corner \(B.\) This will create another rectangle.
Unfold the paper and make a dotted line across the second fold line and label it as \(RS\) as shown.
Square
Again, you notice that the upper side of the line \(RS\) and the lower side are exacly the same/ identical.
Next, fold the paper in half from the bottom left corner \(A\) to the top right corner \(C.\) This creates a triangle.
Unfold the paper, you will notice that a fold line appears along \(BD.\) Trace a dotted line along the fold line as showm.
Square
Finally, fold the paper in half from the top left corner \(D\) to the bottom right corner \(B.\) This creates another triangle.
Now, unfold the paper, you will notice that a fold line appears along \(AC.\) Trace a dotted line along the fold line as showm.
Square
Key Takeaway
There are four dotted lines \(XY, RS, AC \text { and } BD.\) Both sides of each line are exactly the same.
Therefore symmetry is when an object/shape looks exactly similar or identical on one side and the other side when the object is folded/flipped, rotated or reflected.
The four dotted lines \(XY, RS, AC \text { and } BD\) are known are the lines of symmetry. Therefore, a square has \(4\) lines of symmetry.
A line of symmetry divides an object or shape into similar/ identical parts, that is, one half is the mirror image of the other half.

Example 2.2.1.

Find the number of lines of symmetry in the equialteral triangle below.

Solution.

An equilateral triangle has \(3\) lines of symmetry each one from the vertex to the midpoint of the opposite side as shown.

Exercises Exercises

1.
Cut out the following shapes from a paper and find the number of lines of symmetry in each.
2.
Identify lines of symmetry in the alphabetical letters below
alphabets
alphabets-2
alphabets-3
3.
Identify lines of symmetry on the different objects in your classroom and at home.

Subsection 2.2.2 Plane of Symmetry

Activity 2.2.2.

Work in groups
Consider the activity below illustrating plane of symmetry of a cube.
There are \(9\) planes of symmetry in a cube as you can see in the images above.
  • \(2\) vertical planes of symmetry.
  • \(1\) horizontal plane of symmetry.
  • \(6\) diagonal planes of symmetry.
Key Takeaway
Plane of symmetry is an imaginary flat surface that divides an object into two equal halves, such that one half is the mirror image of the other half.
When you cut a banana vertically into two equal halves such that one half is the reflection of the other half, you create a plane of symmetry of the banana.

Example 2.2.2.

How many planes of symmetry does a square based pyramid have?

Solution.

A square based pyramid has \(4\) planes of symmetry as shown below.

Exercises Exercises

1.
How many planes of symmetry are there in the following geometrical objects.
  1. Sphere.
  2. Triangular prism.
  3. Rectangular prism.
  4. Cone.
  5. Cylinder.
2.
Find planes of symmetry for objects around the school.
3.
Identify fruits and vegetables that have planes of symmetry.

Subsection 2.2.3 Properties of Reflection

Activity 2.2.3.

Work in groups
Exploring reflections with mirrors.
Needed Materials
  • A small mirror.
  • Marker or pencil.
  • Piece of paper.
Steps.
  • Hold your hand about \(10 \, cm\) in front of the mirror, with your palm facing the mirror. What do you notice?
    Hand
  • Turn your hand so that the back of your hand faces the mirror instead. Observe again. How does the reflection mimic the shape of your hand? Is there symmetry between your hand and its relection?
  • Hold your hand in front of the mirror with your palm facing the mirror, then fold your fingers to make a fist. Does the reflection mimic your hand movements in the same direction?
  • Slowly rotate your hand in different directions (up, down, sideways) while watching the mirror. Discuss how the mirror’s reflection always mirrors the real movement.
  • Move your hand closer to the mirror until it touches the mirror. What do you notice?
Key Takeaway
  • The mirror does not create a random image but produces a reversed copy of the object in front of it. From your activity, you will notice that when you hold your right hand in front of the mirror, the reflection is a reverse which appers as the left hand. This phenomena is called lateral inversion. Lateral inversion is when a reflected object appears to be flipped along a vertical axis.
  • You will also notice that the reflection mimics you hand movements when you flip, rotate it and when you fold it.
  • As you move your hand closer to the mirror, you notice that the distance between reflection and the mirror reduces. When you touch the mirror, the image appears to touch your hand. Your hand and its reflection are in symmetry.

Exercises Exercises

1.
Identify the parts of your body are the mirror images of one another.
2.
Draw a letter \(E\) on a paper using a marker. Show it to the mirror. Observe how the mirror flips the image.
3.
What other surface reflects images other than a mirror?

Subsection 2.2.4 Reflection of Different Shapes on a Plane

Activity 2.2.4.

Work in groups
Here is a step by step approach on reflection of a triangle on a plane \(M.\)
Draw a perpendicular line from vertex \(A\) to the mirror line \(M\) and measure the distance by counting the number of squares between vertex \(A\) and the mirror line \(M.\) Repeat this process for vertices \(B \text { and }C.\)
Determine the position of the reflected vertices. For vertex \(A\) the perpendicular distance between the vertex and the mirror line is \(2\) squares. Count \(2\) squares from the mirror line to the opposite side of the mirror line and mark that point as \(A',\) which is the reflected image of vertex \(A.\) Repeat the same procedure the remaining vertices \(B \text { and }C.\)
Connect the reflected vertices \(A',B' \text { and }C'\) to create the reflected image of the triangle \(ABC.\)
Reflected triangle

Example 2.2.3.

Draw the image of the pentagon under the reflaction on the diagonal mirror line \(M.\)

Solution.

To obtain the image \(A'\) of \(A\) draw a perpendicular line from \(A\) to the mirro line \(M,\) extend the line the same distance on the opposite side of the mirror line and mark the point as \(A'.\) Similarly, obtain the images \(B', C', D', E'\) the images of vertices \(B, C, D, E\) respectively.
Connect the images of the vertices to form the reflection of the pentagon.
Key Takeaway
  • Reflection moves the image of an object across the mirror line, that is, to the opposite side of the mirror line.
  • A point on the object is the same distance as its reflection from the mirror line.
  • The line connecting a point to its image is perpendicular to the mirror line. Therefore, the mirror line is the perpendicular bisector of the lines connecting the object points and the image points.

Exercises Exercises

1.
Copy the figures below and draw their images under the reflection on the mirror line \(M.\)
2.
Reflect the object about the y-axis
3.
The vertices of a polygon are given as: A(-5,5), B(-6,3), C(-5,1), D(-3,0), E(-2,2) and F(-3,4). Find the image of the polygon under the following reflection lines:
  1. \(y = x\) followed by \(y = 0\)
  2. \(\displaystyle x = 0\)
4.
The points \(A'(-4,1),\,B'(-2,4)\text{ and } C'(-1,3)\) are the images of points \(A,\,B \text{ and } C\) respectively under a reflection on the line \(x = -1.\) Find the coordinates for points \(A,\,B \text{ and } C.\)

Subsection 2.2.5 Determining the Equation of a Mirror Line ( Line of Reflection) Given an Object and its Image

Activity 2.2.5.

Work in groups
Determine the line of reflection that created the reflected image below.
Line of reflection
Copy the figure above on a graph paper
Fold your graph paper such that the points of the objects match with their respective images. Where does the fold line appear?
Key Takeaway
  • You notice that the fold line appears exactly on the y-axis. Therefore, the line of reflection is the y-axis.
  • A line of reflection can be defined with there equation. From the activity, the equation of the line of reflection is \(x = 0.\)

Example 2.2.4.

Determine the equation of the line of reflection.
Line of reflection

Solution.

The coordinates of \(D \text{ and } D'\) are at \((0,0),\) tells you that the line of reflection passes through \((0,0).\)
Connect point \(C \text{ to } C'\) with a line. The line of reflection is the perpendicular bisector of \(C \text{ and } C'.\)
From the properties of reflection, the distance from the object to the mirror line is the same as that of mirror line to the image. Therefore, the line of reflection passes through the midpoint of the line connecting \(C \text{ to } C'.\)
Coordinates for \(C \text{ is }(4,2)\) and that of \(C' \text{ is } (-2,-4).\) The mid point of line \(CC'\) is:
\begin{equation*} \left( \frac{4 + -2}{2},\frac{2 + -4}{2} \right) = (1,-1) \end{equation*}
Since you know that the line of reflection passes through \((0,0) \text{ and } (1,-1),\) the gradient of the reflection line is:
\begin{equation*} m = \frac{-1 - 0}{1 - 0} = -1 \end{equation*}
Therefore taking points \((x,y) \text { and } (1,-1),\) the equation of the line of reflection is :
\begin{equation*} y - y_1 = m (x - x_1) \end{equation*}
\begin{equation*} y - -1 = -1 (x - 1) \end{equation*}
\begin{equation*} y + 1 = -x + 1 \end{equation*}
\begin{equation*} y = -x \end{equation*}

Exercises Exercises

1.
Determine if the transformation is a reflection. If it is a reflection, what is the equation of the mirror line?
Line of reflection
Line of reflection
2.
The vertices of a triangle are \(A(1,2),\, B(3,4)\text{ and }C(5,4).\) The vertices of the image are \(A'(1,-2),\, B'(3,-4)\text{ and }C'(5,-4).\) Find the equation of the line of reflection.
3.
The vertices of a a letter \(V\) are \(P(-3,4),\,Q(-3,2)\text{ and }R(-1,2).\) The vertices of the image are \(P'(-1,2),\, Q'(-1,0)\text{ and }R'(1,0).\) Find the equation of the line of reflection.
4.
\(O(0,0)\) is the centre of a circle of radius \(2\,cm.\) If \(O'(2,0)\) is the reflection of the centre of the circle, find the equation of the line of reflection.

Subsection 2.2.6 Congruence

Activity 2.2.6.

Work in groups
Trace and cut out the following shapes on a piece of paper.
Match two shapes by aligning them directly on top of each other. Discuss how the sides and angles perfectly overlap.
Flip one shape (like turning it over) and try to align it with another shape. Discuss how the shapes still match in size and angles, even though one is flipped.
  • Identify the figures with the same shape.
  • Identify the figures with the same size.
  • Identify figure with same size and shape.
Key Takeaway
  • Figure with the same size and shape are said to be congruent.
  • The symbol for congruence is \(\cong.\) For example, in your activity, \(B \cong I.\)
  • Also, \(B\) fits directly on \(I\) without flipping, thus \(B \text{ and } I\) are said to be directly congruent. Identify other figures that are directly congruent.
  • \(E \text{ and } L\) do not fit directly but when you flip (lateral inversion) figure \(L\) and fit it on \(E,\) they align. This is called opposite congruence. Identify other figure that have opposite congruence from your activity.
  • Figures with the same shape but different size are said to be similar.
  • Congruence can also be applied to line segments and angles.

Example 2.2.5.

Here is an example of congruent polygons.

Exercises Exercises

1.
Identify figures that are:
  • Directly congruent.
  • Oppositely congruent.
  • Similar.
2.
Identify object around your school that have axis of symmetry and state the order of rotational symmetry.

Subsection 2.2.7 Congruence Tests for Triangles

Activity 2.2.7.

Work in groups
Conditions for Congruence in Triangles
Materials
  • Construction paper
  • Pencil, ruler, protractor
Instructions
Trace the following triangles on a construction paper.
Identify pairs of congruent triangles.
From the pairs of congruent triangles you have identified, which pairs fit the following criteria:
  1. The three sides of one triangle is equal to the three sides of the corresponding triangle.
  2. Two sides and an included angle of one triangle is equal to the two corresponding sides and the included angle of the other triangle.
  3. One side and two included angles of one triangle is equal to the corresponding side and the two included angles of the other triangle.
  4. One side and the hypotenuse of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle.
Key Takeaway
Congruence in triangles dependS on the measure of the sides and angles. Two triangles are said to be congruent if a pair of the corresponding sides and corresponding angles are equal.
Criteria for congruence tests in triangles include:
  • Side-side-side (SSS): the three sides of one triangle is equal to the three sides of the corresponding triangle.
  • Side-angle-side (SAS): two sides and an included angle of one triangle is equal to the two corresponding sides and the included angle of the other triangle.
  • Angle-side-angle (ASA): one side and two included angles of one triangle is equal to the corresponding side and the two included angles of the other triangle.
  • Right angle-hypotenuse-side (RHS): one side and the hypotenuse of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle.
  • Angle-angle-side (AAS): one side and two included angles of one triangle is equal to the corresponding side and the two included angles of the other triangle.

Example 2.2.6.

Check if the triangles below are congruent and state the test of congruence criterion.

Solution.

From the figure, identify corresponding sides and angles.
Sides \(AB = PR = 3\, cm \text{ and } BC = PQ = 8\,cm\)
\(\angle B = \angle P = 60^\circ.\)
Therefore, \(\Delta ABC \cong \Delta PQR\) by SAS criterion

Exercises Exercises

1.
Check if the triangles below are congruent and state test of congruence criterion.
2.
Show that \(\Delta ABC \cong \Delta ADB\) if \(AD = AE = BE = BC.\)
3.
\(A(0,4)\, B(-3,0) \text{ and } C(0,2)\) are the coordinates of \(\Delta ABC.\) Reflect the triangle over mirror line \(x = 0.\) Prove that the triangle and its image are congruent and state the test of congruence criterion.
4.
Construct an equilateral triangle \(UVW\) with sides \(6\,cm.\,X\) is the midpoint of \(UW\) and \(VX\) is perpendicular to \(UW.\) Show that \(\Delta UVX \, \cong \, \Delta VWX.\) State the test of congruence criterion.
PrevTopNext