<em><u>Statement:</u></em>
The hypotenuse of a right angled triangle is 2√13 cm. If the smaller side is increased by 2 cm and the larger side is increased by 3 cm, the new hypotenuse will be √117 cm.
<em><u>To find out:</u></em>
The length of the larger side of the right angled triangle.
<em><u>Solution:</u></em>
Let us consider x as the smaller side and y as the larger side.
Then, in the right angled triangle,
x² + y² = (2√13)² ...(I) [By Pythagoras Theorem]
Now, if the smaller side is increased by 2 cm, then the smaller side will be (x + 2).
And if the larger side is increased by 3 cm, then the larger side will be (y + 3).
Then, in the new right angled triangle,
(x + 2)² + (y + 3)² = (√117)² [By Pythagoras Theorem]
or, x² + 2 × 2 × x + 2² + y² + 2 × 3 × y + 3² = (√117)²
or, x² + 4x + 4 + y² + 6y + 9 = (√117)²
or, x² + y² + 4x + 6y + 13 = (√117)²
Now, put the value of x² + y² from equation (I),
or, (2√13)² + 4x + 6y + 13 = (√117)²
or, (2 × 2 × √13 × √13) + 4x + 6y + 13 = (√117 × √117)
or, 52 + 4x + 6y + 13 = 117
or, 4x + 6y = 117 - 52 - 13
or, 4x + 6y = 52
or, 4x = 52 - 6y
or, x = ...(II)
Now, put the value of x of equation (II) in (I),
x² + y² = (2√13)²
or, + y² = 52
or, = 52
or, 52y²-624y + 2704 = 52 × 16
or, 52y² - 624y + 2704 - 832 = 0
or, 52y² - 624y + 1872 = 0
or, 52(y² - 12y + 36) = 0
or, y²-12y +36 = 0 ÷ 52
or, y²-12y +36 = 0
or, (y)² - 2 × 6 × y + (6)² = 0
or, (y - 6)² = 0
or, y - 6=0
or, y = 6
We have taken y as the length of the larger side of the right angled triangle.
So, the length of the larger side is 6 cm.
<em><u>Answer:</u></em>
6 cm
Hope you could understand.
If you have any query, feel free to ask.
