Question

In $\vartriangle ABC$, $\angle A = {90^0}$, AB= 5 cm and AC = 12 cm. If $AD \bot BC$, then AD=
(1) $\dfrac{{60}}{{13}}$cm
(2) $\dfrac{{11}}{2}$cm
(3) $\dfrac{{13}}{{60}}$cm
(4) $\dfrac{{2\sqrt {15} }}{{13}}$cm

Answer 1

Hint: Here we will use the law of similarity for two triangles. This law will establish the ratio of corresponding sides. This helps to find the length of AD. Also make use of the pythagoras theorem to find the lengths of the sides.

Complete step-by-step answer:

In the above diagram first we will compute the length of BC by using Pythagoras theorem.
AB= 5 cm and AC = 12 cm
BC = ?
According to the Pythagoras theorem for right angled triangle ABC, we have,
\[
  B{C^2} = A{B^2} + A{C^2} \\
   \Rightarrow B{C^2} = {5^2} + {12^2} \\
   \Rightarrow BC = \sqrt {169} \\
   \Rightarrow BC = 13cm \\
 \]
Now in $\vartriangle ABC and \vartriangle ADC$
$\angle C$ is a common angle.
And $\angle A = \angle D = {90^0}$ (AD is perpendicular to BC)
Therefore, $\vartriangle ABC \sim \vartriangle ADC$ means both triangles are similar by AA similarity criterion.
Therefore, the property of similar triangles ratio of the corresponding sides of the two triangles will be equal. Thus in other words, we can write,
$
  \dfrac{{AD}}{{AB}} = \dfrac{{AC}}{{BC}} \\
   \Rightarrow AD = \dfrac{{AB \times AC}}{{BC}} \\
   \Rightarrow AD = \dfrac{{12 \times 5}}{{13}} \\
   \Rightarrow AD = \dfrac{{60}}{{13}} \\
 $(Transforming by algebra rules and putting the known values.)
Thus the length of AD will be $\dfrac{{60}}{{13}}$cm.
Therefore, the correct answer is option (1).

Note: This question is a simple application of the law of similarity in the triangles. Furthermore many problems of geometry can be solved by using these laws. Not only can this one find the missing angles of any triangle. One interesting fact about the similar triangles that their areas are also in the ratio of their altitudes also.