Q16 In an equilateral triangle, prove that three times the square of one side is equal to four times - Maths - Triangles

Question

Q16 In an equilateral triangle, prove that three times the square
of one side is equal to four times the square of one of its
altitudes

Lovina Kansal · Accepted Answer

Dear student
16)
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Given: Equilateral triangle ABC with each side a and AD as one of its altitudesTo prove: 3×Square of one side=4×Square of one of it's altitude⇒3a2=4AD2Proof:In △ADB and △ADCAB=AC   Both are a as it is equilateral △AD=AD   (common)∠ADB=∠ADC   Each 90°Hence △ADB≅△ADC   By R
H
S congruencyHence, BD=DC    CPCT⇒BD=DC=12BC⇒BD=DC=a2     
(1)Now, ADB is a right-angled △Using Pythagoras theoremAB2=AD2+BD2⇒a2=AD2+a22    using 1⇒AD2=a2-a24⇒AD2=3a24⇒4AD2=3a2Hence proved

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Q16. In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.