The hypotenuse of a right triangle is 3√10 cm. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be 9√5 cm. How long are the legs of the triangle?
Answers
SOLUTION :
Given : Hypotenuse of a right triangle is 3√10 cm.
Let the length of smaller leg be x cm and the larger leg be y cm.
By using Pythagoras theorem,
H² = B² + P²
x² + y² = (3√10)²
x² + y² = 9×10
x² + y² = 90
y² = 90 - x² . ………….(1)
Given if the smaller leg is tripled and the longer leg is doubled, new hypotenuse will be 9√5 cm
Smaller leg = 3x
longer leg = 2y
By using Pythagoras theorem,
H² = B² + P²
(3x)² + (2y)² = (9√5)²
9x² + 4y² = 81×5
9x² + 4y² = 405
9x² + 4(90 - x²) = 405
[From eq 1]
9x² + 360 - 4x² = 405
9x² - 4x² + 360 - 405 = 0
5x² - 45 = 0
5x² = 45
x² = 45/5
x² = 9
x = √9
x = ±3
Since, length of a side can't be negative, so x ≠ - 3
Therefore, x = 3
Length of smaller leg be 3 cm
On putting x = 3 in eq 1,
y² = 90 - x²
y² = 90 - 3²
y² = 90 - 9
y² = 81
y = √81
y = ± 9
Since, length of a side can't be negative, so y ≠ - 9
Therefore, y = 9
Length of larger leg be 9 cm
Hence, the length of smaller leg be 3 cm and the larger leg be 9 cm.
HOPE THIS ANSWER WILL HELP YOU…
Hii dear here is your answer
Let the Sides of the Triangle be a,b and c.
By the Pythagoras Theorem
a^2+b^2=c^2
Given relation- a^2+b^2= (3√10)^2=90……(1)
Now, smaller leg, i.e 'a' is tripled, thus new side is 3a
The other leg is doubled, i.e 2b.
Now given relation,
(3a)^2+(2b)^2 = (9√5)^2
9a^2+4b^2= 405……….(2)
By eq.(1) and eq.(2)
There are 2 variables and 2 equations.Thus, they can be solved by many methods, the simplest ones being Substitution or Elimination.
We will use Substitution.
Eq.1 can also be written as → a^2=90-b^2
Substituting this value in the second equation
9(90-b^2) + 4b^2= 405
810–9b^2 + 4b^2= 405
Thus -5 b^2 = -405
b^2= 81
b=9cm
Thus a^2 + 9^2 = 90
a^2= 90–81=9
Thus a=3 cm.
Hope its help you ✔✔