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Theorem In a Right-Angled Triangle with sides in A.P

Piyush, born on 10th Feb, 1967, Aquarian belongs to a middle class family in Dadri, Near Noida, elder son of Dr. Devender Kumar Goel and mother  Ravikanta. He is Diploma Mechanical Engineering passed in the year 1987, Diploma in Material Management, Diploma in Vastu Shastra and Diploma in Business Management. creative, believe in God too much, believe in Love & Friendship.

Piyush is now known as ‘Mirror Image ManPiyush is very passionate about mathematics he has done lot work in Mathematics .
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Theorem: In a Right-Angled Triangle with sides in A.P by Piyush Goel
Theorem: In a Right-Angled Triangle with sides in A.P. Series, the distance between the point of intersection of median & altitude at the base is 1/10th the sum of other two sides.

This Theorem applies in Two Conditions:

  1. The Triangle must be Right-Angled.
  2. Its Sides are in A.P. Series.

We Have:

triangle-theoram

  1. ∆ABC is Right-Angled
  2. AD is Altitude
  3. AE is Median i.e. E is the midpoint of BC

Proof:

(a+d)2 = a2 + (a-d)2

(a+d)2 -(a-d)2 = a2

a2 + d2 + 2ad – a2 – d2 + 2ad = a2

4ad = a2

a(a-4d) = 0

a – 4d = 0 (as a ≠ 0)

a = 4d (———-eqn. 1)

 

In ∆ABD

AB2 = BD2 + AD2

(a – d)2 = BD2 + AD2

(a – d)2 = {(a + d)/2 – DE}2 + AD2 (———-eqn. 2)

In ∆ACD

AC2 = DC2 + AD2

a2 = DC2 + AD2

a2 = {(a + d)/2 + DE}2 + AD2 (———-eqn. 3)

 

From eqn. 2 & 3, we get

(a – d)2 – a2 = {(a + d)/2 – DE}2 + AD2 – {(a + d)/2 + DE}2 – AD2

(a – d +a )(a – d – a) = {(a+d)/2 – DE + (a+d)/2 + DE}{(a+d)/2 – DE – (a+d)/2 – DE}

(2a – d)(-d) = (a + d)(-2DE)

(2a – d)(d) = (a + d)(2DE)

So, 2DE = (2a – d)d/(a+d)

From eqn. 1, we get

2DE = (2*4d – d)d/(4d + d)

2DE = 7d2/5d

DE = 7d/10 = (4d + 3d)/10

But, AB = a –d = 4d – d = 3d & AC = a = 4d

Putting these values, we get

DE = (AC + AB)/10 (Hence Proved)

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Devang Vibhakar is the Founder and Editor of www.SpeakBindas.com. He has interviewed more than 350 people. His effort was recognized by Limca Book of Records, twice. He has been to Scotland as well as Germany as part of vocational & cultural exchange programs and has compiled five books so far. He's passionate about bringing forth interesting stories & interviews of entrepreneurs to avid readers of SpeakBindas. He can be reached here.

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About

Devang Vibhakar
Devang Vibhakar
(Editor, www.SpeakBindas.com)
SpeakBindas is a Limca Book of Awards winning blog. It's a PR3 blog with 1000+ articles & Interviews. Devang Vibhakar - Founder - has interviewed more than 350+ people and has been to Scotland, London & Germany. He's a writer, speaker, traveler & movie-goer !! He has also written four books. He's passionate about bringing forth interesting stories & interviews of people from various sectors of life to avid readers of SpeakBindas. You can reach him here .