In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3BC. Prove that 9AD^2 = 7AB^2.
Triangles
1
Posts
2
Users
0
Likes
180
Views
0
06/06/2021 12:16 pm
Topic starter
In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3BC. Prove that 9AD^{2} = 7AB^{2}.
Answer
Add a comment
Add a comment
1 Answer
0
06/06/2021 12:22 pm
ABC is an equilateral triangle.
And D is a point on side BC such that BD = 1/3BC
Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.
∴ BE = EC = BC/2 = a/2
And, AE = a√3/2
Given, BD = 1/3BC
∴ BD = a/3
DE = BE – BD = a/2 – a/3 = a/6
In ΔADE, by Pythagoras theorem,
AD^{2} = AE^{2} + DE^{2 }
AD^{2} = \((\frac{a \sqrt{3}}{2})^2 + (\frac{a}{6})^2\)
= \((\frac{3a^2}{4} + (\frac{a^2}{36})\)
= \(\frac{28a^2}{36}\)
= \(\frac{7}{6}AB^2\)
⇒ 9 AD^{2} = 7 AB^{2}
Add a comment
Add a comment
Forum Jump:
Related Topics

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod.
3 years ago

In Figure, D is a point on side BC of ∆ ABC such that BD/CD = AB/AC. Prove that AD is the bisector of ∠BAC.
3 years ago

In Figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that: (i) ∆ PAC ~ ∆ PDB (ii) PA.PB = PC.PD.
3 years ago

In Figure, two chords AB and CD intersect each other at the point P. Prove that : (i) ∆APC ~ ∆ DPB (ii) AP.PB = CP.DP
3 years ago

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
3 years ago
Forum Information
 321 Forums
 27.3 K Topics
 53.8 K Posts
 0 Online
 12.4 K Members
Our newest member: Stripchat
Forum Icons:
Forum contains no unread posts
Forum contains unread posts
Topic Icons:
Not Replied
Replied
Active
Hot
Sticky
Unapproved
Solved
Private
Closed