Triangles that cut off similar triangles, and the SSS and SAS tests
Answers are for parents and teachers.
1
DE ∥ BC with D on AB and E on AC: AD = 5, DB = 3, DE = 10, AE = 7.5. Find BC and AC.
AB = 8 and AD/AB = 5/8. BC = 10 × 8/5 = 16 and AC = 7.5 × 8/5 = 12.
2
In a right triangle at A with AD ⊥ BC: BD = 9 and DC = 16. Find AD, BC, AB and AC.
AD² = 144, AD = 12. BC = 25. AB² = 25 × 9 = 225, AB = 15. AC² = 25 × 16 = 400, AC = 20. (15² + 20² = 25² ✓.)
3
Are the triangles with sides 7, 8, 12 and 21, 24, 36 similar? Give k.
7/21 = 8/24 = 12/36 = 1/3: yes, by SSS with k = 1/3.
4
△ABC: AB = 10, AC = 14, ∠A = 60°. △DEF: DE = 5, DF = 7, ∠D = 60°. Are they similar?
10/5 = 14/7 = 2 with the equal angles between these sides: yes, by SAS with k = 2.
5
In △ABC, D is on BC with AC = 6, CB = 12 and CD = 3. Prove △ACD ~ △BCA.
AC/CB = 6/12 = 1/2 and CD/AC = 3/6 = 1/2, and ∠C is common (between these sides). So △ACD ~ △BCA by SAS.