Question

If A be the area of a right triangle and b is one of the sides containing the right angle. Find m if the length of the altitude on the hypotenuse is $\dfrac{{{\text{mAb}}}}{{\sqrt {{{\text{b}}^4}{\text{ + 4}}{{\text{A}}^2}} }}$.

Answer 1

Hint: To solve this question, we will first find the area of the triangle by using the formula A = ½ x base x height. Then, we will find the hypotenuse with the help of Pythagoras theorem. Then, by using hypotenuse and the given length of altitude, we will make another equation and compare it to the equation made by using formula to find the area of the triangle.

Complete step-by-step answer:
Now, we say we have a right triangle ABC, with the right triangle at C. The length of base is given which is b and let the length of perpendicular be x. Now, we will draw the figure according to questions.
 

In the above figure, AD is the hypotenuse, BC is the perpendicular and BC = x units. AC is the base and AC = b units. DC is altitude from vertex C on the hypotenuse and its length is $\dfrac{{{\text{mAb}}}}{{\sqrt {{{\text{b}}^4}{\text{ + 4}}{{\text{A}}^2}} }}$.
Now, Area of a right triangle is found by the formula A = $\dfrac{1}{2}$(base) x (height)
According to the question, the area of triangle ABC is A and A = $\dfrac{1}{2}$(b)(x)
$ \Rightarrow $ \[{\text{x = }}\dfrac{{2{\text{A}}}}{{\text{b}}}\] ……. (1)
Now, we will use the Pythagorean theorem to find the length of the hypotenuse. Now, according to Pythagorean theorem the square of length of hypotenuse is equal to the sum of squares of length of base and perpendicular. So, applying Pythagoras theorem in triangle ABC, we get
Length of hypotenuse = $\sqrt {{{\text{x}}^2}{\text{ + }}{{\text{b}}^2}} $
Now, putting value of x from equation (1) in the above equation, we get
Length of hypotenuse = $\sqrt {{{\left( {\dfrac{{2{\text{A}}}}{{\text{b}}}} \right)}^2}{\text{ + }}{{\text{b}}^2}} $ = $\dfrac{{\sqrt {{\text{4}}{{\text{A}}^2}{\text{ + }}{{\text{b}}^4}} }}{{\text{b}}}$ …… (2)
Now, area of triangle can be written as
A = $\dfrac{1}{2}$(length of hypotenuse) x (length of altitude in hypotenuse)
So, from equation (2), we get
A = $\dfrac{1}{2}$ x $\dfrac{{\sqrt {{\text{4}}{{\text{A}}^2}{\text{ + }}{{\text{b}}^4}} }}{{\text{b}}}$ x $\dfrac{{{\text{mAb}}}}{{\sqrt {{{\text{b}}^4}{\text{ + 4}}{{\text{A}}^2}} }}$
A = $\dfrac{1}{2}$ x $\dfrac{{{\text{mAb}}}}{{\text{b}}}$
On solving, we get 2A = mA.
So, the value of m is 2.

Note: when we come up with such types of questions, we will follow a few steps to solve the problem. First, we will find the area of the triangle by applying the formula A = $\dfrac{1}{2}$(base) x (height). Then we will apply Pythagoras theorem to find the length of the hypotenuse. The length is always positive, so we have taken positive signs when we removed under – root from hypotenuse. Then, we will again apply the formula of the area to find the value of the asked variable.