Question

the hypotenuse of a right triangle is 3√5 cm. if the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15cm. find the length of each side. ​

Answer 1

Answer:

3 cm and 6 cm

Step-by-step explanation:

Let length of smaller side be x cm and larger side be y cm.

By using Pythagoras theorem,

x² + y²  = (3√5)²

x² + y² = 45

y² = 45 - x² . ………….(1)

If the smaller side is tripled and the longer side is doubled, new hypotenuse will be 15 cm.

Smaller side = 3x  

longer side = 2y

Apply Pythagoras theorem,

(3x)² + (2y)² = (15)²

9x² + 4y²  = 225

9x² + 4(45 - x²) = 225

9x² + 180 - 4x² = 225

9x² - 4x² + 180 - 225 = 0

5x² - 45 = 0

x² = 9

x = ±3

Since, length of a side can't be negative, so x ≠ - 3  

Hence, x = 3

Thus, Length of  smaller side = 3 cm

Place x = 3 in (1),

y² = 45 - x²

y² = 45 - 3²

y² = 36

y = ± 6

Since, length of a side can't be negative, so y ≠ -6  

Therefore, y = 6

Length of larger side be 6 cm

Hence, the length of  smaller side be 3 cm and the larger leg be 6 cm.

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Answer 2

Let smaller side = x CM

Larger side = y CM

Hypotenuse = 3√5cm

By Pythagoras theorem

$\longrightarrow$$\sf\green{ {x}^{2} + {y}^{2} = {(3 \sqrt{5}) }^{2} }$

$\longrightarrow$$\sf\green{{x}^{2} + {y}^{2} = 45}$

$\longrightarrow$$\sf\green{{y}^{2} = 45 - {x}^{2} }$

When the smaller side is tripled and the larger side is doubled, the hypotenuse becomes 15cm.

$\therefore$ $\sf\red{( {3x}^{2} ) - ( {2y)}^{2} = {15}^{2}}$

$\longrightarrow$$\sf\red{ {9x}^{2} + 4(45 - {x)}^{2} = 225}$

$\longrightarrow$$\sf\red{ {5x}^{2} = 225 - 180 = 45}$

$\longrightarrow$$\sf\red{ {x}^{2} = 9}$

$\longrightarrow$$\sf\red{x = ±3}$

As the side of a triangle cannot be negative,

x ≠ -3 , so x = 3

Hence, the length of the smaller side us 3cm and the length of the larger side is 6cm.