What does it mean?
Similar figures keep their shape while all corresponding lengths share one scale factor. Angles stay equal. The direction matters: if a small side of 4 cm corresponds to a large side of 12 cm, the small-to-large factor is 3, while the reverse factor is 1/3.
An enlargement (dilation) from a centre multiplies every length by a scale factor k > 0. The image is similar to the preimage: corresponding angles are equal and corresponding sides are in the same ratio k. Congruence is the special case k = 1 after a rigid motion. AA: if two pairs of corresponding angles are equal, the triangles are similar. To find a missing length, match corresponding sides, then multiply or divide by k. The diagram is not to scale. Use marked angles and given lengths only.
Learning goal: Understand similarity through dilation; use scale factor for lengths. The grade is a suggested starting point; school sequences vary. Use the first worked example to check readiness, and Grade 6 ratio and unitary method: word problems explained if an earlier idea needs revisiting.
See the idea
Read each diagram with its labels, then follow the worked examples. Practice questions include their own measurements and information.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Choose a method and explain why it works
Match corresponding vertices using angles or the named order. Divide image length by original length to find k. Multiply every original corresponding length by k to find the image. For triangles, two equal pairs of angles are enough for AA similarity because their third angles must then agree too.
Notation: △ABC ∼ △DEF. Scale factor k = image length ÷ matching original length.
Key rule and when to use it
Similarity preserves corresponding angles and gives one common length scale factor. With positive scale k, areas scale by k² and volumes of similar solids by k³.
Worked examples, step by step
Example 1: A side of 4 cm is enlarged by scale factor 3 from a centre. Find the image length.
- Corresponding lengths multiply by k.
- Image length = 3 × 4 = 12 cm.
- Angles stay the same. Only lengths change.
Answer: 12 cm
Check: Three copies of 4 cm fit the image side.
Example 2: Triangles have ∠A = ∠D = 50° and ∠B = ∠E = 70°. A side opposite A is 6 cm. The corresponding side opposite D is unknown. k from ABC to DEF is 2. Find that side, after naming a similarity reason.
- Two pairs of angles are equal, so AA. The triangles are similar.
- Similarity means corresponding sides scale by k = 2.
- The side corresponding to 6 cm is 12 cm. Do not scale a side that is not corresponding.
Answer: AA; 12 cm
Check: k = 2 doubles the matching side, not a random side.
Example 3: A 6 cm by 4 cm rectangle is enlarged by scale factor 2.5. Find its dimensions and area.
- New lengths are 6 × 2.5 = 15 cm and 4 × 2.5 = 10 cm.
- The new area is 150 cm². The old area was 24 cm².
- Check area scaling: 24 × 2.5² = 24 × 6.25 = 150 cm².
Common mistakes and important boundaries
A reduction has 0 < k < 1 in this guide’s positive-scale models. Lengths scale by k, areas by k² and volumes of similar solids by k³. Multiplying an area by just k is a common mismatch between the kind of measurement and the scale factor.
A point to explain
A learner may treat similar as congruent, scale a non-corresponding side, or infer equality from a sketch.
Compare an uncertain attempt with the closest worked example. Look at the meaning of each operation, unit and label before repeating the arithmetic.
Similar Shapes and Scale Factor — practice
Work on paper first. Open each answer afterwards and compare the reasoning. Print the questions with or without the separate answer key; no account is required.
Name: ____________________ Date: __________
What is true of similar triangles?
Answer and a way to check
Corresponding angles are equal and corresponding sides are proportional.
Similar means same angles, sides in ratio k. Same size is k = 1, which is also congruent.
Two triangles each have angles 50° and 70°. Why are the triangles similar?
Answer and a way to check
Two pairs of angles are equal, so AA.
Name the two equal angle pairs. AA does not need a side.
A length of 5 cm is enlarged by k = 4. Find the image length.
Answer and a way to check
20 cm
Image length = k times the matching original length.
An image side is 18 cm. The matching original side is 6 cm. Find the scale factor from original to image.
Answer and a way to check
3
Divide the image length by the matching original length.
A reduction uses k = 1/2. An original side is 14 cm. Find the image length.
Answer and a way to check
7 cm
Multiply 14 by one half. Do not double.
Triangles are similar by AA. A student multiplies a side by k but uses a side that is not corresponding. Why is the length wrong?
Answer and a way to check
k compares matching sides between the two triangles. A non-corresponding side belongs to a different pair.
Match vertices in order. Then match the sides between those vertices.
k from small to large is 2. A student says the triangles are congruent because the sketch looks the same. What is the mistake?
Answer and a way to check
k = 2 means lengths double, so they are similar but not the same size.
Congruent is k = 1 after a rigid motion. k = 2 changes lengths.
A photo is 8 cm by 6 cm. An enlargement is similar with k = 1.5. Find both image sides.
Answer and a way to check
12 cm by 9 cm
Multiply each given side by 1.5. Do not add 1.5.
Similar Shapes and Scale Factor — answer key
Corresponding angles are equal and corresponding sides are proportional. Similar means same angles, sides in ratio k. Same size is k = 1, which is also congruent.
Two pairs of angles are equal, so AA. Name the two equal angle pairs. AA does not need a side.
20 cm Image length = k times the matching original length.
3 Divide the image length by the matching original length.
7 cm Multiply 14 by one half. Do not double.
k compares matching sides between the two triangles. A non-corresponding side belongs to a different pair. Match vertices in order. Then match the sides between those vertices.
k = 2 means lengths double, so they are similar but not the same size. Congruent is k = 1 after a rigid motion. k = 2 changes lengths.
12 cm by 9 cm Multiply each given side by 1.5. Do not add 1.5.
Questions students and parents ask
Can similar triangles be congruent?
Yes. When the corresponding length scale factor is 1, their matching sides have equal lengths, so they are congruent as well as similar.
Can two rectangles fail to be similar?
Yes. They need the same ratio of corresponding side lengths. A 2-by-4 rectangle and a 2-by-6 rectangle both have right angles but different proportions.
Is this a complete set of textbook exercise solutions?
No. This guide adapts BananaLearn’s own teaching examples and practice. It is not official NCERT material or a promise of the exact chapter sequence in a particular board, country or textbook edition.
What if the examples are still difficult?
Revisit Grade 6 ratio and unitary method: word problems explained. Try the first example with its model, then attempt one fresh question independently. A correct copied answer is not evidence that the method can be used without support.
Help at home
Ask which side corresponds before calculating. A correct multiplication applied to the wrong pair of sides still produces an incorrect geometric answer.
Keep one example of the learner’s own working. A repeated difficulty across fresh questions can guide a discussion with their teacher; one short practice sheet does not establish a diagnosis or a full mastery profile.
Connect this idea to the next skill
About this guide
Based on BananaLearn’s Grade 8 course lesson “Same Shape, Scale the Lengths”. The explanations have been adapted for independent reading; course diagrams are reused where they fit the idea, and missing classroom givens are made explicit in questions.
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