PLZ GUYS HELP IM DOOMED WITH THIS QUES.
Answers
Answer:
(i) The sides are 8cm, 15cm and 17cm.
(ii) The sides are 9cm, 40cm and 41cm.
Step-by-step explanation:
(i)
We have,
In △ABC,
∠B = 90°
AB = (x - 3) cm
BC = (x + 4) cm
AC = (x + 6) cm
Now,
We must know that,
In a triangle, that is, a triangle with one of the Angles equal to 90°, it must follow the Pythagorus theorem.
But we must be very careful with sides,
So, Draw a triangle with the above given measurements,
Then, apply the Pythagorus theorem,
(Side)² + (Side)² = (Hypotenuse)²
Hypotenuse is the side with longest length,
Here out of all the sides,
(x + 6) will be the greatest
So,
AB² + BC² = AC²
(x - 3)² + (x + 4)² = (x + 6)²
Using the identity,
(a + b)² = a² + 2ab + b²
And
(a - b)² = a² - 2ab + b²
We get,
(x² - 6x + 3²) + (x² + 8x + 4²) = (x² + 12x + 6²)
(x² - 6x + 9) + (x² + 8x + 16) = (x² + 12x + 36)
x² - 6x + 9 + x² + 8x + 16 = x² + 12x + 36
2x² + 2x + 25 = x² + 12x + 36
2x² - x² + 2x - 12x + 25 - 36 = 0
x² - 10x - 11 = 0
Now, we must factorise
Sum = b = (-10)
Product = a × c = (-11)
So, Factors are (-10) and 1
Then,
x² - 11x + x - 11 = 0
x(x - 11) + 1(x - 11) = 0
(x + 1)(x - 11) = 0
So,
x = (-1) or x = 11
Both could be a result,
but for AB = x - 3
If we put x = (-1) we will get a negative value
And we know that sides can't be negative
So,
x = 11
Thus, Sides are
AB = x - 3 = 11 - 3 = 8 cm
BC = x + 4 = 11 + 4 = 15 cm
AC = x + 6 = 11 + 6 = 17 cm
Hence,
The sides are 8cm, 15cm and 17cm.
(ii)
Same procedure as above,
We have,
In △ABC,
∠B = 90°
AB = x cm
BC = (4x + 4) cm
AC = (4x + 5) cm
Now,
We must know that,
In a triangle, that is, a triangle with one of the Angles equal to 90°, it must follow the Pythagorus theorem.
But we must be very careful with sides,
So, Draw a triangle with the above given measurements,
Then, apply the Pythagorus theorem,
(Side)² + (Side)² = (Hypotenuse)²
Hypotenuse is the side with longest length,
Here out of all the sides,
(4x + 5) will be the greatest
So,
AB² + BC² = AC²
(x)² + (4x + 4)² = (4x + 5)²
Using the identity,
(a + b)² = a² + 2ab + b²
We get,
(x²) + (16x² + 32x + 4²) = (16x² + 40x + 5²)
x² + (16x² + 32x + 16) = (16x² + 40x + 25)
x² + 16x² + 32x + 16 = 16x² + 40x + 25
x² + 16x² - 16x² + 32x - 40x + 16 - 25 = 0
x² - 8x - 9 = 0
Now, we must factorise
Sum = b = (-8)
Product = a × c = (-9)
So, Factors are (-9) and 1
Then,
x² - 9x + x - 9 = 0
x(x - 9) + 1(x - 9) = 0
(x + 1)(x - 9) = 0
So,
x = (-1) or x = 9
Both could be a result,
but for BC = 4x + 4
If we put x = (-1) we will get 0
And we know that sides can't be 0
So,
x = 9
Thus, Sides are
AB = x = 9cm
BC = 4x + 4 = 4(9) + 4 = 36 + 4 = 40cm
AC = 4x + 5 = 4(9) + 5 = 36 + 5 = 41cm
Hence,
The sides are 9cm, 40cm and 41cm.
Hope it helped you and believing you understood it...All the best