The worksheets themselves are in German – just as in class. The editor is only available in German so far.
Similarity & intercept theorems – worksheets to print
Enlargement and scale factor, missing sides of similar triangles, the intercept theorems in V and X figures, similar or not, and word problems such as shadows, the pinhole camera and ramps.
- Choose the number of tasks and the difficulty yourself
- With an answer sheet and, if you like, a QR code to the answers
- At three levels (★ to ★★★) and as group A and B
- As a single worksheet or as a task on a mixed worksheet
Similarity & intercept theorems: explained simply
Similar figures have the same shape but a different size: every length is stretched by the same factor k. The intercept theorems use this to work out missing lengths.
How to work it out
- Scale factor k = image length : original length. k > 1 enlarges, k < 1 reduces.
- Areas change with k²: with k = 2 an area becomes four times as large.
- First intercept theorem: SA : SB = SC : SD.
- Second intercept theorem: SA : SB = AC : BD when AC and BD are parallel.
An example, step by step
Two rays from S are cut by the parallel lines AC and BD. SA = 4 cm, SB = 6 cm, AC = 3 cm. How long is BD?
- k = SB : SA = 6 : 4 = 1.5.
- By the second intercept theorem, BD = AC · k.
- BD = 3 cm · 1.5 = 4.5 cm.
Understand other variations
Similar triangles
ABC has a = 4 cm, b = 5 cm, c = 6 cm. A′B′C′ is similar with a′ = 8 cm. Then k = 2, b′ = 10 cm and c′ = 12 cm.
Similar or not?
4, 5, 6 cm and 6, 7.5, 9 cm: 6 : 4 = 1.5, 7.5 : 5 = 1.5 and 9 : 6 = 1.5 – similar. Sort the sides by size first.
X figure
If S lies between the parallel lines, the same holds: SA : SB = SC : SD = AC : BD. A and B are then on different sides of S.
Shadows
A 1.5 m stick casts a 2 m shadow, a tree at the same time a 12 m shadow. Height of the tree: 1.5 m · 12 : 2 = 9 m.
Your turn
k = 0.5. A line segment is 7 cm long. How long is its image?
Show the answer and method
7 cm · 0.5 = 3.5 cm.
Ready-made worksheets: Similarity & intercept theorems
Click the picture to see the worksheet large and print it. “Use” opens it in the editor.