Geometry

Intercept theorems and similarity

When two lines through one point are cut by parallels, the pieces are in the same ratio. München names the intercept theorems (Strahlensätze) for its T-Kurs test.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• First theorem: the segments on one ray are in the same ratio as the matching segments on the other ray.
• Second theorem: the parallel segments are in the same ratio as the segments from the centre.
• Similar figures: all lengths scale by the same factor k, areas by k², volumes by k³.
• Set up the ratio with matching sides, then cross-multiply.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Two triangles are similar. In the small one, side A = 5 and side B = 4. In the large one, side A = 10. How long is side B in the large one?

   • a8
   • b9
   • c10
   • d6
   Show solution

   Answer: 8

   Similar figures: all corresponding sides share the same scale factor.

   Scale factor = 10 / 5 = 2. So side B = 4 · 2 = 8.

2. 2. Two triangles are similar. In the small one, side A = 4 and side B = 3. In the large one, side A = 12. How long is side B in the large one?

   • a10
   • b9
   • c12
   • d6
   Show solution

   Answer: 9

   Similar figures: all corresponding sides share the same scale factor.

   Scale factor = 12 / 4 = 3. So side B = 3 · 3 = 9.

3. 3. Two triangles are similar. In the small one, side A = 5 and side B = 8. In the large one, side A = 10. How long is side B in the large one?

   • a8
   • b10
   • c17
   • d16
   Show solution

   Answer: 16

   Similar figures: all corresponding sides share the same scale factor.

   Scale factor = 10 / 5 = 2. So side B = 8 · 2 = 16.

4. 4. Two triangles are similar. In the small one, side A = 6 and side B = 5. In the large one, side A = 12. How long is side B in the large one?

   • a7
   • b10
   • c12
   • d11
   Show solution

   Answer: 10

   Similar figures: all corresponding sides share the same scale factor.

   Scale factor = 12 / 6 = 2. So side B = 5 · 2 = 10.

5. 5. Two triangles are similar. In the small one, side A = 8 and side B = 7. In the large one, side A = 24. How long is side B in the large one?

   • a24
   • b10
   • c22
   • d21
   Show solution

   Answer: 21

   Similar figures: all corresponding sides share the same scale factor.

   Scale factor = 24 / 8 = 3. So side B = 7 · 3 = 21.

6. 6. Two triangles are similar. In the small one, side A = 3 and side B = 4. In the large one, side A = 6. How long is side B in the large one?

   • a8
   • b4
   • c6
   • d9
   Show solution

   Answer: 8

   Similar figures: all corresponding sides share the same scale factor.

   Scale factor = 6 / 3 = 2. So side B = 4 · 2 = 8.

Common mistakes

• Using a piece of a ray when the theorem needs the whole segment from the centre.
• Scaling areas by k instead of k².
• Pairing sides that do not correspond.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs