Question

The sides of an equilateral triangle are shortened by 12 units 13 units and 14 units respectively and a right angle triangle is formed. The side of the equilateral triangle

Answer 1

Let the side of equilateral triangle be x.

The sides are shortened by 12 , 13 , 14 units respectively.

The new sides are ( x - 12 ) , ( x - 13 ) , ( x - 14)

Given that,

A right angled triangle is formed .

( x - 13)² + ( x - 14)² = ( x - 12)²

x² + 169 + x² + 196 -26x - 28x = x² + 144 - 24x

x² + 25 + 196 = 54x - 24x

x² + 221 = 30x

x² - 30 x + 221 = 0

x² - 13x - 17x + 221 = 0

x ( x - 13 ) - 17 ( x - 13 ) = 0

( x - 17 ) ( x - 13 ) =  0

x = 17 or 13

If x = 13 , then sides of right angle are 1 , 0 , -1 which isn't possible.

So, x = 17 .

Therefore, Side of the equilateral triangle is 17cm .

Answer 2

Let x be the length of side of the equilateral side. 

Given : The sides of an equilateral triangle are shortened by 12 units, 13 units, 14 units and a right angle triangle is formed.  

Then, new lengths of the triangle  are

➞(x - 12), (x - 13) and (x - 14)

These are the sides of a right angle triangle and (x - 12) is the longest side. 

Because the longest side of the right triangle is hypotenuse, (x - 12) can be considered to be the length of the hypotenuse. 

According to Pythagorean theorem, the square of the hypotenuse is equal to sum of the squares of other two sides.

Then, we have ,  

⇒(x - 13)² + (x - 14)²  =  (x - 12)²

⇒x² - 2(x)(13) + 132 + x2 - 2(x)(14) + 142  =  x² - 2(x)(12) + 144

⇒x² - 26x + 169 + x² - 28x + 196  =  x² - 24x + 144

⇒2x² - 54x + 365  =  x² - 24x + 144

⇒ x² - 30x + 221  =  0

⇒(x - 13)(x - 17)  =  0

⇒x  =  13  or  x  =  17

If we take x  =  13, the sides of the right triangle are

x - 12  =  1,   x - 13  =  0,  x - 14  =  -1

When x  =  13, we get one of the sides is zero and the sign of the another side is negative.  

Then, x  = 13 can not be accepted.

If we take x  =  17, the sides of the right triangle are

➞x - 12  =  5,   x - 13  =  4,  x - 14  =  3

All of the three sides of the right angle are positive when x  =  17.

Moreover, the lengths 5, 4 and 3 satisfy the Pythagorean theorem.  

That is,

⇒52  =  42 + 32

⇒25  =  16 + 9

⇒25  =  25

Therefore, x=17 can be accepted. 

So, the length of side of equilateral triangle is 17 units