Congruence & Sequences of Transformations — Challenge
Two shapes are congruent exactly when some sequence of slides, flips and turns carries one onto the other.
What these problems 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
Which of these is not a valid test for two triangles being congruent?
- SSS
- SAS
- ASA
- AAA
- RHS
Worked answer
AAA
SSS, SAS, ASA and RHS each fix the size of the triangle as well as its shape.
Knowing all three angles fixes only the shape: a small triangle and a large one can have identical angles.
So AAA proves similarity but not congruence.
Problem 2
Triangle ABC is congruent to triangle DEF, and AB=5. What is the length of DE?
- 2.5
- 5
- 10
- 25
- It cannot be determined
Worked answer
5
In a congruence statement the letters are listed in matching order, so AB corresponds to DE.
Congruent triangles have equal corresponding sides, so DE = AB = 5.
Problem 3
One triangle has sides 3, 4, 5 and another has sides 6, 8, 10. How are they related?
- Congruent
- Similar but not congruent
- Neither similar nor congruent
- Both congruent and similar
- It cannot be determined
Worked answer
Similar but not congruent
Each side of the second triangle is exactly twice the matching side of the first, so the shapes match but the sizes do not.
Equal ratios of corresponding sides means the triangles are similar.
They are not congruent, because congruence would require the sides to be equal rather than proportional.
Problem 4
Congruent figures have the same what?
- Shape only
- Size only
- Size and shape
- Area only
- Perimeter only
Worked answer
Size and shape
Congruent figures can be placed exactly on top of one another.
That requires both the same shape and the same size.
Equal area alone is not enough: a long thin rectangle and a square can share an area without matching.
Problem 5
Triangle ABC is congruent to triangle XYZ, and angle A is 40 degrees. How many degrees is angle X?
- 40
- 50
- 80
- 140
- 100
Worked answer
40
The order of the letters pairs A with X.
Corresponding angles of congruent triangles are equal, so angle X is 40 degrees.
Problem 6
Which transformation always produces an image congruent to the original?
- Reflection
- Dilation with factor 3
- Dilation with factor 0.5
- A horizontal stretch
- A shear
Worked answer
Reflection
A reflection flips the figure but changes no distance between its points.
Since all lengths and angles survive, the image is congruent to the original.
Both dilations change every length, and stretches and shears distort the shape.
Problem 7
A triangle is congruent to one with sides 7, 9 and 12. What is its perimeter?
- 24
- 26
- 28
- 30
- 84
Worked answer
28
Congruent triangles have the same three side lengths.
So the perimeter is 7+9+12 = 28.
Multiplying the sides gives 756, and 84 comes from 7· 12; neither is a perimeter.
Problem 8
If two triangles are congruent, how do their areas compare?
- They are equal
- They are in the ratio 2:1
- They are in the ratio 1:2
- They are different
- It cannot be determined
Worked answer
They are equal
Congruent triangles have identical sides and angles, so one can be laid exactly over the other.
Anything measured from those sides and angles must therefore agree, including the area.
So the areas are equal.