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Ex.4.2 Q5 Quadratic Equations Solutions - NCERT Maths Class 10

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Question

The altitude of right triangle is \(7\,\rm{ cm}\) less than its base. If the hypotenuse is \(13\,\rm{ cm},\) find the other two sides.

 Video Solution
Quadratic Equations
Ex 4.2 | Question 5

Text Solution

What is known?

i) Altitude of right triangle is \(7\, \rm{cm}\) less than its base.

ii) Hypotenuse is \(13\, \rm{cm}\)

What is Unknown?

The measure of the two sides of a given right triangle.

Reasoning:

In a right triangle, altitude is one of the sides. Let the base be \(x\, \rm{cm.}\)

The altitude will be \((x - 7)\, \rm{cm.}\)

Next, we can apply the Pythagoras theorem to the given right triangle.

Pythagoras theorem:

\(\text{Hypotenuse}^2= \text{Side 1}^2+ \text{Side 2}^2\)

\[13{}^\text{2}\text{ }=\text{ }x{}^\text{2}\text{ }+\text{ }(x\text{ }-7)\text{ }{}^\text{2}\]

Steps:

\[\begin{align}13&= x + {(x - 7)^2}\\169 &= x \!\!+ \!\!x -\!\! 14x  \!\!+ \!\!49  \\169 &= 2x - 14x + 49 \end{align}\] 

\[\begin{align} 2x \!\!- \!\!14x \!\!+ \!49 \!- \!169  & \!=\! 0\\  2x - 14x \\ - 120  &= 0\\  \frac{{ 2x - 14x - 120 }}{2}&= 0\\x - 7x-60 &= 0\\
 x\! - \!12x \!+\! 5x \! - \! 60  &= 0\\  x \! \left( \! x - 12 \! \right) \!+ \! 5\! \left( {x - 12} \right)  & = 0\\(x + 5)(x - 12) &= 0\\ \end{align}\]

\[ \begin{align} x-12 = 0 &\qquad x + 5 = 0\\x = 12 &\qquad \quad\;\; x = - 5\end{align}\]

We know that the value of the base cannot be negative. 

\(\therefore\) Base \(= x = 12 \,\rm{cm}\)

Altitude \(= 12 -7 = 5 \,\rm{cm}\)

Lengths of two sides are \(12\, \rm{cm}\) and \(5\,\rm{ cm.}\)

  
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