Congruence & Sequences of Transformations — Practice
Two shapes are congruent exactly when some sequence of slides, flips and turns carries one onto the other.
What these questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Corresponding sides — Sides in matching positions in two similar figures.
- Perimeter — Distance all the way around the outside.
- Rigid motion — A move that keeps every length and every angle. Slides, flips and turns are all rigid.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Orientation — Whether the labels run clockwise or counter-clockwise around the shape.
- Dilation — A resizing from a centre point by a scale factor k.
- Transformation — A change to the function that moves or stretches its graph.
- Sequence — An ordered list of numbers. The nth one is an.
Problem 1
△ ABC ≅ △ DEF and AB = 7. Find DE.
Worked answer
7
AB corresponds to DE
Corresponding sides are equal
DE = 7
Problem 2
△ PQR ≅ △ XYZ and ∠ Q = 55°. Find ∠ Y.
Worked answer
55
∠ Q corresponds to ∠ Y
∠ Y = 55°
Problem 3
△ LMN ≅ △ RST with MN = 12. Which side of △ RST is 12?
Worked answer
ST
MN corresponds to ST
ST = 12
Problem 4
A square is rotated 45°. What is its new side length if the original was 6 cm?
Worked answer
6
Rotation changes position, not size
The side is still 6 cm
Problem 5
Which sequence guarantees the image is congruent to the original?
- Reflect, then translate
- Dilate by 2, then translate
- Reflect, then dilate by 3
- Dilate by 12, then rotate
Worked answer
Reflect, then translate
Reflections and translations are both rigid
Dilations are not, unless k = 1
So the first sequence is the only one
Problem 6
A(4, 1) is reflected across the x-axis and then translated 3 left. Find the image.
Worked answer
(1, -1)
Reflect: (4, 1) → (4, -1)
Translate 3 left: (1, -1)
Problem 7
B(-2, 5) is translated 6 right and then reflected across the y-axis. Find the image.
Worked answer
(-4, 5)
Translate: (-2, 5) → (4, 5)
Reflect across the y-axis: (-4, 5)
Problem 8
△ ABC ≅ △ DEF with AB = 3x - 4 and DE = 2x + 5. Find x.
Worked answer
9
3x - 4 = 2x + 5
x = 9
Check: 3(9) - 4 = 23 and 2(9) + 5 = 23 ✓
Problem 9
△ GHI ≅ △ JKL with ∠ G = (4y + 10)° and ∠ J = (6y - 14)°. Find y.
Worked answer
12
4y + 10 = 6y - 14
24 = 2y
y = 12
Problem 10
A shape is reflected twice, across two different lines. Is the final orientation the same as the original or reversed?
Worked answer
same
One reflection reverses it
A second reflection reverses it back
So the orientation is the same
Problem 11
A figure and its image have opposite orientations. Which sequence could have produced it?
- A reflection followed by a translation
- Two rotations
- A translation followed by a rotation
- Three rotations
Worked answer
A reflection followed by a translation
Rotations and translations keep the orientation
An odd number of reflections reverses it
So the answer must contain exactly one reflection
Problem 12
△ ABC ≅ △ DEF. The perimeter of △ ABC is 31. Find the perimeter of △ DEF.
Worked answer
31
Every corresponding side is equal
So the perimeters are equal
31
Problem 13
P(1, 4) is rotated 180° about the origin and then translated by (x + 5, y - 2). Find the image.
Worked answer
(4, -6)
Rotate: (1, 4) → (-1, -4)
Translate: (-1 + 5, -4 - 2) = (4, -6)
Problem 14
△ ABC ≅ △ DEF. AB = 5, BC = 8 and the perimeter of △ DEF is 20. Find AC.
Worked answer
7
Perimeter of △ ABC is also 20
5 + 8 + AC = 20
AC = 7
Problem 15
Which single transformation has the same effect as reflecting across the x-axis and then across the y-axis?
Worked answer
a 180 degree rotation about the origin
(3, 5) → (3, -5) → (-3, -5)
Both coordinates ended up negated
That is the rule for a 180° rotation about the origin
Problem 16
△ ABC ≅ △ DEF, ∠ A = 40° and ∠ E = 75°. Find ∠ C.
Worked answer
65
∠ B = ∠ E = 75°
∠ C = 180 - 40 - 75
∠ C = 65°