Question

ABC is an equilateral triangle the altitude AD is 5√3 find its sides hint call all the sides as 'a' use pythogoras theorem ​

Answer 1

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Answer

Length of side is 10cm

Given

• Length of altitude = 5√3cm

To Calculate

• Sides of triangle

Solution

Let the side be a

So, DC = 1/2 a

Now, ADC is a right angle △

In △ADC

By Pythagorous Theorem

$\bf {Hypotenuse}^{2} = {Perpendicular}^{2} + {Base}^{2}$

$\bf \implies AD {}^{2} + DC {}^{2} = AC {}^{2}$

$\bf \implies {(5 \sqrt{3} )}^{2} + ( \frac{1}{2} a) {}^{2} = {a}^{2}$

$\bf \implies 75 + \frac{1}{4} {a}^{2} = {a}^{2}$

$\bf \implies \frac{1}{4} (300 + {a}^{2} ) = {a}^{2} ──eq.1$

In △ ADB

$\bf \implies AD {}^{2} + DB {}^{2} = AB {}^{2}$

$\bf \implies \frac{1}{4} (300 + {a}^{2} ) = {a}^{2} ──eq.2$

By adding Equation 1 and 2

$\bf \implies\frac{1}{4} (300 + {a}^{2} ) + \frac{1}{4} (300 + {a}^{2} ) = {a}^{2} + {a}^{2}$

$\bf \implies75 + \frac{1}{4} {a}^{2} + 75 + \frac{1}{4} {a}^{2} = {a}^{2} + {a}^{2}$

$\bf \implies150 + \frac{1}{2} {a}^{2} = 2 {a}^{2}$

$\bf \implies150 = {2a}^{2} - \frac{1}{2} {a}^{2}$

$\bf \implies150 = \frac{3}{2} {a}^{2}$

$\bf \implies \frac{3}{2} {a}^{2} = 150$

$\bf \implies {a}^{2} = 150 \times \frac{2}{3}$

$\bf \implies {a}^{2} = 50 \times 2$

$\bf \implies {a}^{2} = 100$

$\bf \implies a = \sqrt{100}$

$\bf \implies a = 10$

Therefore, side of triangle is 10cm.