Question

State and prove Pythagoras theorem. Using this theorem, find the altitude of

an equilateral triangle ABC whose side is 2a.​

Answer 1

Pythagoras theorem statement:-

“In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“

$\mapsto \underline{\boxed{\sf Hypotenuse^2= Perpendicular^2+ Base^2}}$

Proof:-

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\put(0,0){\line(1,0){4.5}}\put(0,0){\line(0,1){5}}\qbezier(0,5)(0,5)(4.5,0)\qbezier(0,0)(0,0)(2.3,2.48)\put(-0.5,5.2){\bf C}\put(-0.5,-0.5){\bf B}\put(4.7,-0.5){\bf A}\put(2.5,2.6){\bf D}\end{picture}$

Let,  ABC is a right angled triangle,right-angled at B

Therefore,

• AC = hypotenuse
• AB = adjacent side
• BC = perpendicular

To Prove :

• AC²= AB² + BC²

Construction: Draw a perpendicular BD meeting AC at D

From the diagram,

→ΔADB ~ ΔABC  

As the corresponding sides of similar triangles are proportional,

⇒AD/AB = AB/AC

⇒AB²= AD × AC -------(1)

Also,

→ΔBDC ~ΔABC

As the corresponding sides of similar triangles are proportional,

⇒CD/BC = BC/AC

⇒ BC²= CD × AC -------(2)

Adding the equations (1) and (2),

⇒AB² + BC²= AD × AC+ CD × AC

⇒AB²+ BC² = AC (AD + CD)

We know,

• AD + CD = AC

Therefore,

$\\\mapsto \underline{\boxed{\sf AB^2 + BC^2 = AC^2 }}$

Hence proved !

---------------------------

Altitude of an equilateral triangle whose side is 2a:-

Given:

• ABC is an equilateral triangle
• Side = 2a

To find:

• Altitude

Solution:

Let ABC be a equilateral triangle .

Therefore,AB=BC=AC=2a

AD is perpendicular on BC.

We know,

• Altitude of an equilateral triangle bisects opposite side

Hence,

→BD=CD=a

In ΔADB,  

We have:

• AB = hypotenuse
• AD = perpendicular
• BC = base

By Pythagoras theorem,

$:\implies \sf AB^2 = AD^2 +BD^2\\\\:\implies \sf (2a)^2 = AD^2 + BD^2 \\\\:\implies \sf 4a^2 = AD^2 + a^2\\\\:\implies \sf 4a^2 - a^2 = AD^2\\\\:\implies \sf 3a^2 = AD^2 \\\\ :\implies \sf AD =\sqrt{3}a\\\\ :\implies \boxed{\boxed{\bold{Altitude = \sqrt{3}a.}}}$

Hence,

Length of altitude is √3a.

______________

Answer 2

Step-by-step explanation:

it is ok for you if yes please give thanks