D Is the midpoint of hypotenuse …

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D Is the midpoint of hypotenuse AC of a right angle triangle ABC. prove that BD=1/2 AC.

Posted by Gulshan Sinku 4 years ago

CBSE > Class 09 > Mathematics

1 answers

My Guide ... 4 years ago

Let ∆ABC is a right triangle such that ∠B = 90 and D is mid point of AC . then we have to prove that BD = 1/2 AC we produce BD to E such that BD = AC and EC. Now is ∆ADB and ∆CDE we have AD = DC [Given] BD = DE [By construction] And, ∠ADB = ∠CDE [Vertically opposite angles] ∴ By SAS criterion of congruence we have ∆ADB ≅ ∆CDE ⇒ EC = AB and ∠CED = ∠ABD ....(i) [By cpctc] But ∠CED & ∠ABD are alternate interior angles ∴ CE ║ AB ⇒ ∠ABC + ∠ECB = 1800 [Consecutive interior angles] ⇒ 90 + ∠ECB = 1800 ⇒ ∠ECB = 900 Now, In ∆ABC & ∆ECB we have AB = EC [By (i)] BC = BC [Common] And, ∠ABC = ∠ECB = 900 ∴ BY SAS criterion of congruence ∆ABC ≅ ∆ECB ⇒ AC = EB [By cpctc] ⇒ 1/2 AC = 1/2 EB ⇒ BD = 1/2 AC

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