Question

prove that three times the square of any side of an
equilateral trangle is equal to four times the square of the
alitude ?​

Answer 1

$\large\underline{\sf{Solution-}}$

Let us assume triangle ABC as equilateral triangle with AB = BC = CA.

Let further assume that AD be the altitude drop from vertex A intersecting BC at D.

Let assume that,

AB = BC = CA = x and AD = y

Thus,

We have to prove that

$\rm :\longmapsto\:\green{\boxed{ \bf{ \: {3x}^{2} = {4y}^{2}}}}$

Now, Consider,

$\bf:\longmapsto\:In \: \triangle \: ADC \: and \: \triangle \: ADB$

$\rm :\longmapsto\:AD \: = \: AD \: \: \: \: \: \: \{common \}$

$\rm :\longmapsto\:AC \: = \: AB \: \: \: \: \{equal \: sides \}$

$\rm :\longmapsto\:\angle \: ADB \: = \: \angle \: ADC \: \: \: \: \{each \: 90 \degree \}$

$\bf:\longmapsto\: \: \triangle \: ADC \: \cong \: \triangle \: ADB \: \: \: \: \{RHS \}$

$\rm :\longmapsto\:DB \: = \: DC \: \: \: \: \{CPCT \}$

$\bf\implies \:DB \: = \: \dfrac{x}{2}$

Now,

$\rm :\longmapsto\:In \: \triangle \: ADB$

Using Pythagoras Theorem,

$\rm :\longmapsto\: {AB}^{2} = {AD}^{2} + {DB}^{2}$

$\rm :\longmapsto\: {x}^{2} = {y}^{2} + {\bigg(\dfrac{x}{2} \bigg) }^{2}$

$\rm :\longmapsto\: {x}^{2} = {y}^{2} + \dfrac{ {x}^{2} }{4}$

$\rm :\longmapsto\: {4x}^{2} = {4y}^{2} + {x}^{2}$

$\rm :\longmapsto\: {4x}^{2} - {x}^{2} = {4y}^{2}$

$\rm :\longmapsto\: {3x}^{2} = {4y}^{2}$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

1. Pythagoras Theorem :-

• This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

• This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

• This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem,

• If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.