Question

If D is the midpoint of hypotenuse AC of right triangle ABC, prove that BD=1/2 triangle ABC

Answer 1

Given : ΔABC in which ∠B = 90 0 and D is the mid point of AC.

Prove that : BD = ½ AC.

Construction : Produce BD to E so that BD = DE. Join EC.

Statements and Reasons

1) AD = DC
1) Given

2) BD = DE
2) By construction

3) ∠ADB = ∠CDE
3) Vertically opposite angles

4) ΔADB ≅ ΔCDE
4) By SAS postulate

5) EC = AB and ∠CED = ∠ABD
5) CPCTC

6) CE || AB
6) If alternate interior angles are congruent then the lines are parallel

7) ∠ABC + ∠ECB = 180
7) Angles formed on the same side of transveral are supplementary.

8) 90 + ∠ECB = 180
8) Since ∠B = 90 given

9) ∠ECB = 180 -90 = 90
9) By subtraction property

10) AB = EC
10) From (5)

11) BC = CB
11) Reflexive (Common side)

12) ∠ABC = ∠ECB
12) Each 900

13) ΔABC ≅ ΔECB
13) SAS postulate

14) AC = BE
14) CPCTC

15) 1/2AC = 1/2BE ⇒ 1/2AC = BD
15) Multiply by 1/2 but 1/2BE = BD by mid point definition.