Congruent triangles · Ages 13–14
Congruent figures and transformations
Translate, rotate, reflect: congruent; enlarge or reduce: not.
How to do it
Method
Translate, rotate, reflect: congruent; enlarge or reduce: not
Sides 3 and 4 enlarged twice become 6 and 8: not congruent
Another way
Tracing paper
Slid, turned or flipped, it covers the other exactly: congruent
In full
Two figures with the same shape and the same size are congruent. A translation, a rotation or a reflection only changes where a figure is or which way it faces, never its size or shape, so the image is congruent to the figure: corresponding sides equal, corresponding angles equal. An enlargement or a reduction is different: the shape stays but the lengths change, so it is not congruent (it is similar). One way: measure corresponding sides. Count each side's length on the grid and compare them one by one; one different means not congruent. Another: imagine tracing one figure and sliding, turning or flipping the tracing paper: does it cover the other exactly? Careful: a reflected figure looks back to front, but it is still congruent.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
△A′B′C′ is the image of △ABC after a transformation, as shown, and the two are congruent. A′B′ is units long.
Show the answer and working
Answer: 3
Corresponding sides of congruent figures are equal: A′B′ = AB = 3 units.
Example 2
After which of these transformations is the image not congruent to the figure?
- A. a translation
- B. a rotation
- C. a reflection
- D. an enlargement, 3 times
Choose the right answer
Show the answer and working
Answer: D. an enlargement, 3 times
Enlarged 3 times, every side becomes 3 times as long: the same shape, a different size, so not congruent.
Example 3
△ABC turned 90° about the origin gives △A′B′C′, as shown. A′B′ = units, A′C′ = units, and the area of △A′B′C′ = square units.
Fill in each blank
Show the answer and working
Answer: 2, 4, 4
The turned image is congruent: A′B′ = AB = 2, A′C′ = AC = 4, and its area = 2 × 4 ÷ 2 = 4.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The blue triangle is the image of the orange one after a transformation. The two triangles are congruent.
True or false?
Answer:
2.△A′B′C′ is the image of △ABC after a transformation, as shown, and the two are congruent. A′B′ is units long.
Answer:
3.A square of side 8 cm is transformed, and its image has sides of 9 cm. The transformation could be a rotation.
True or false?
Answer:
4.△ABC is reflected to give △A′B′C′, then △A′B′C′ is enlarged 4 times to give △A″B″C″. AB = 8 cm, the perimeter of △ABC is 17 cm, and ∠B = 50°. A″B″ = cm, the perimeter of △A″B″C″ = cm, and ∠B″ = °.
Fill in each blank
Answer:
5.△ABC turned 90° about the origin gives △A′B′C′, as shown. A′B′ = units, A′C′ = units, and the area of △A′B′C′ = square units.
Fill in each blank
Answer:
All the answers
- 1. True
- 2. 4
- 3. False
- 4. 32, 68, 50
- 5. 4, 2, 4
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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