Question

The altitude of a right-angled triangle is 7cm less than its base. If the hypotenuse is 13cm, find the other two sides.

Answer 1

Hint: We use Pythagoras theorem to solve questions when right angled triangles are given. Pythagoras theorem is stated as: In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

Complete Step-by-Step solution:
We are given that the altitude of a right-angled triangle is 7cm less than its base and the hypotenuse is 13cm. We have to find the other two sides.
We first draw the figure of the given problem and then proceed to solve it.
Let the triangle be ABC where the hypotenuse is side AC, base is BC and the height of the triangle is AB.
Let the base side BC be x
Then, according to the given conditions in the question we have the height AB as x-7.
Then the figure is obtained as,

Now because the given triangle is a right angled triangle then by using the Pythagoras theorem, we will get the value of x.
Pythagoras theorem says ‘In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.’
Using Pythagoras theorem in triangle ABC we have,
\[A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\]
Substituting the value of AC, AB and BC in above we get,
\[\begin{array}{*{35}{l}}
   {{13}^{2}}=\text{ }{{\left( x-7 \right)}^{2}}+\text{ }{{\left( x \right)}^{2}} \\
   \begin{align}
  & \Rightarrow 169\text{ }=\text{ }{{x}^{2}}\text{+ }49\text{ }\text{ }14x~+\text{ }{{x}^{2}} \\
 & \Rightarrow 169\text{ - }49\text{ }=\text{ }2{{x}^{2}}\text{ }14x \\
\end{align} \\
   \Rightarrow 2{{x}^{2}}-14x-120=0 \\
   {} \\
\end{array}\]
By splitting middle term in the above expression we have,
\[\begin{align}
  & \Rightarrow {{x}^{2}}-7x-60=0 \\
 & \Rightarrow {{x}^{2}}-12x+5x-60=0 \\
 & \Rightarrow x(x-12)+5(x-12)=0 \\
 & \Rightarrow (x-12)(x+5)=0 \\
\end{align}\]
\[\Rightarrow x=12,x=-5\]
Now, x=-5 is not possible because the side of a triangle cannot be negative, therefore x=12 is the answer.
Hence, the base of the triangle is BC = 12cm and the height of the triangle AB is
 x-7 = 12-7 = 5cm
Therefore, the other two sides of the triangle are 12cm and 5cm.

Note: To solve this type of question we always assume one of the sides as some variable, say x, then we proceed according to the given situations in the question to get the value of both the sides.