# The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the product of the lengths of these sides.

Answer

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Hint: We need to use Pythagoras theorem to solve this problem as hypotenuse and other two sides are given.

Let one side of the given triangle be $x$ and the other side becomes $x + 5$.

From Pythagoras theorem, $${\left( {hypotenuse} \right)^2} = {(adj.side)^2} + {(opp.side)^2}$$

$ \Rightarrow {(25)^2} = {x^2} + {\left( {x + 5} \right)^2}$

$ \Rightarrow 625 = {x^2} + {x^2} + 10x + 25$

$ \Rightarrow 2{x^2} + 10x - 600 = 0$

$$ \Rightarrow {x^2} + 5x - 300 = 0$$

$ \Rightarrow {x^2} + 20x - 15x - 300 = 0$

$$ \Rightarrow x(x + 20) - 15(x + 20) = 0$$

$$ \Rightarrow \left( {x + 20} \right)\left( {x - 15} \right) = 0$$

$ \Rightarrow (x + 20) = 0$ $\& $$\left( {x - 15} \right) = 0$

$ \Rightarrow x = - 20\& x = 15$

But $x$ (length of a side) cannot be negative.

Hence one side is $x = 15$cm

Other side is $x + 5 = 15 + 5 = 20$cm

The product of the length of sides $ = 15 \times 20 = 300c{m^2}$

Note: In a right triangle, the hypotenuse is the longest side, an opposite side is the one across from a given angle, and an adjacent side is next to a given angle. Pythagoras theorem provides us with the relation between the sides in a right triangle. A right triangle consists of two legs and a hypotenuse. The two legs meet at an angle of ${90^ \circ }$. The hypotenuse is the longest side of the right triangle and is the side opposite to the right angle. Pythagoras theorem states that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.

Let one side of the given triangle be $x$ and the other side becomes $x + 5$.

From Pythagoras theorem, $${\left( {hypotenuse} \right)^2} = {(adj.side)^2} + {(opp.side)^2}$$

$ \Rightarrow {(25)^2} = {x^2} + {\left( {x + 5} \right)^2}$

$ \Rightarrow 625 = {x^2} + {x^2} + 10x + 25$

$ \Rightarrow 2{x^2} + 10x - 600 = 0$

$$ \Rightarrow {x^2} + 5x - 300 = 0$$

$ \Rightarrow {x^2} + 20x - 15x - 300 = 0$

$$ \Rightarrow x(x + 20) - 15(x + 20) = 0$$

$$ \Rightarrow \left( {x + 20} \right)\left( {x - 15} \right) = 0$$

$ \Rightarrow (x + 20) = 0$ $\& $$\left( {x - 15} \right) = 0$

$ \Rightarrow x = - 20\& x = 15$

But $x$ (length of a side) cannot be negative.

Hence one side is $x = 15$cm

Other side is $x + 5 = 15 + 5 = 20$cm

The product of the length of sides $ = 15 \times 20 = 300c{m^2}$

Note: In a right triangle, the hypotenuse is the longest side, an opposite side is the one across from a given angle, and an adjacent side is next to a given angle. Pythagoras theorem provides us with the relation between the sides in a right triangle. A right triangle consists of two legs and a hypotenuse. The two legs meet at an angle of ${90^ \circ }$. The hypotenuse is the longest side of the right triangle and is the side opposite to the right angle. Pythagoras theorem states that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.

Last updated date: 26th Sep 2023

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