☰ Year plan · Mathematics

Triangles that cut off similar triangles, and the SSS and SAS tests

Week 6 · Similar triangles: the parallel-line corollary, the altitude to the hypotenuse, and the SSS and SAS cases · 3 lessons
2

The altitude to the hypotenuse of a right triangle: three similar triangles and the relations (DA)² = DB × DC and (AB)² = BC × BD

Mathematics Sec 1 Term 1, Unit 2, Lesson 2
GoalProve and use the relations between the sides of a right triangle and the altitude to its hypotenuse.
New words
hypotenuse · the longest side of a right triangle, opposite the right anglealtitude · the perpendicular from a vertex to the opposite sidemean proportional · x is the mean proportional of a and b when x² = a × b
Explanation

In △ABC with a right angle at A, draw AD perpendicular to BC. The altitude cuts the big triangle into two smaller right triangles, and all three are similar. △DBA and △ABC share the angle B and each has a right angle, so △DBA ~ △ABC. △DAC and △ABC share the angle C and each has a right angle, so △DAC ~ △ABC. Hence △DBA ~ △DAC. Writing the proportions of matching sides gives three facts: DA/DC = DB/DA, so (DA)² = DB × DC (the altitude is the mean proportional of the two parts of the hypotenuse); AB/BC = BD/AB, so (AB)² = BC × BD; and AC/BC = CD/AC, so (AC)² = CB × CD. Adding the last two gives AB² + AC² = BC(BD + CD) = BC², which is Pythagoras’ theorem again. Remember: (DA)² = DB × DC, not DA = DB × DC.

ABCD
Worked examples
In a right triangle at A with AD ⊥ BC: BD = 4 and DC = 9. Then (AD)² = 4 × 9 = 36, AD = 6. BC = 13. (AB)² = BC × BD = 13 × 4 = 52, AB = 2√13. (AC)² = CB × CD = 13 × 9 = 117, AC = 3√13. Check: 52 + 117 = 169 = 13² ✓.
In the same kind of triangle AD = 12 and BD : DC = 4 : 9. Let BD = 4k and DC = 9k. Then 36k² = 144, k = 2, so BD = 8 and DC = 18, BC = 26. AB² = 26 × 8 = 208, AB = 4√13; AC² = 26 × 18 = 468, AC = 6√13. Check: 208 + 468 = 676 = 26² ✓.
AB = 10 and BC = 25. Then AB² = BC × BD gives 100 = 25 × BD, so BD = 4 and DC = 21. AD² = 4 × 21 = 84, and AC² = 25 × 21 = 525. Check: 100 + 525 = 625 = 25² ✓.
Class activity

Cut and compare. Draw a right triangle with legs 6 cm and 8 cm. Draw the altitude to the hypotenuse, measure BD, DC and AD with a ruler, and check (AD)² = BD × DC.

Practice
1
BD = 3 and DC = 12. Find AD.
2
BC = 20 and BD = 5. Find AB, DC, AD and AC.
3
The legs are AB = 6 and AC = 8. Find BC, BD, DC and AD.
4
Which segment is the mean proportional between BD and DC?
5
Explain why △DBA ~ △ABC.