Question

In the figure, triangle ABC is right-angled at B. Given that AB=9cm, AC=15cm, and D, E are the mid-points of the sides AB and AC respectively. Calculate, the length of BC.

Answer 1

Hint: The given triangle is a right-angle triangle which is right-angled at B. We know that we can apply Pythagoras theorem in a right-angle triangle. We can apply Pythagoras theorem in the \[\Delta ABC\] and solve it further.

Complete step-by-step solution -

According to the figure, we have \[\Delta ABC\] which is right-angled at B.
We know that we can apply Pythagoras theorem in a right-angle triangle.
We have AB as the height of the triangle and it measures 9cm.
We have AC as the hypotenuse of the triangle and it measures 15cm.
Now, we can apply Pythagoras theorem.
\[{{\left( Hypotenuse \right)}^{2}}={{\left( Height \right)}^{2}}+{{\left( Base \right)}^{2}}\] …………………………(1)
On solving, we can write equation (1) as
\[{{\left( Base \right)}^{2}}={{\left( Hypotenuse \right)}^{2}}-{{\left( Height \right)}^{2}}\] ……………………..(2)
Putting, Hypotenuse=15cm and Height=9cm in equation (2), we get
\[{{\left( Base \right)}^{2}}={{\left( 15 \right)}^{2}}-{{\left( 9 \right)}^{2}}\] ………….(3)
We know that, \[{{15}^{2}}=225\] and \[{{9}^{2}}=81\] .
Putting \[{{15}^{2}}=225\] and \[{{9}^{2}}=81\] in equation (3), we get
\[\begin{align}
  & {{\left( Base \right)}^{2}}={{\left( 15 \right)}^{2}}-{{\left( 9 \right)}^{2}} \\
 & \Rightarrow {{\left( Base \right)}^{2}}=225-81 \\
 & \Rightarrow {{\left( Base \right)}^{2}}=144 \\
 & \Rightarrow Base=12 \\
\end{align}\]
In \[\Delta ABC\] , we have BC as base and after applying Pythagoras theorem in the \[\Delta ABC\] , we have got base which is equal to 12cm.
Hence, the length BC in \[\Delta ABC\] is equal to 12cm.

Note: We have solved this question only by using the Pythagoras theorem. So, we have only used the height and hypotenuse. Since we have not used the remaining data provided in the question. So, one may easily get confused about what to do with the remaining data provided. So, don’t get confused and keep your focus only to find the base. The remaining data has nothing to do and is meaningless in the process to find the base.