The length of a hypotenuse of a right-angled triangle exceeds the length of its base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle (in cm).
(A) 6, 8, 10
(B) 7, 24, 25
(C) 8, 15, 17
(D) 7, 40, 41
VerifiedHint: Assume that the length of the hypotenuse of the right-angled triangle is \[x\] cm. Since the length of its base is 2 cm less than its hypotenuse so, the length of the base of the given right-angled triangle is \[\left( x-2 \right)\] cm. Since the altitude is 1 cm less than half of the length of the hypotenuse so, the length of the altitude of the given right-angled triangle is \[\left( \dfrac{x-1}{2} \right)\] cm. Now, use Pythagoras theorem, \[{{\left( Hypotenuse \right)}^{2}}={{\left( Altitude \right)}^{2}}+{{\left( Base \right)}^{2}}\] and calculate the value of x.
Complete step by step answer:
According to the question, we are given that the length of a hypotenuse of a right-angled triangle exceeds the length of its base by 2 cm and exceeds twice the length of the altitude by 1 cm.
First of all, let us assume that the length of the hypotenuse of the right-angled triangle is \[x\] cm ………………………………………….(1)
It is given that the length of its base is 2 cm less than its hypotenuse ………………………………………(2)
Now, from equation (1) and equation (2), we get
The length of the base of the given right-angled triangle = \[\left( x-2 \right)\] cm ……………………………….(3)
It is also given that the altitude is 1 cm less than half of the length of the hypotenuse …………………………………………..(4)
Now, from equation (1) and equation (4), we get
The length of the altitude of the given right-angled triangle = \[\left( \dfrac{x-1}{2} \right)\] cm ……………………………………(5)
We know the Pythagoras theorem, \[{{\left( Hypotenuse \right)}^{2}}={{\left( Altitude \right)}^{2}}+{{\left( Base \right)}^{2}}\] ……………………………………….(6)
Now, from equation (1), equation (3), equation (5), and equation (6), we get
$ \Rightarrow {{\left( x \right)}^{2}}={{\left( x-2 \right)}^{2}}+{{\left( \dfrac{x-1}{2} \right)}^{2}} $
$ \Rightarrow {{x}^{2}}={{x}^{2}}+4-4x+\dfrac{{{x}^{2}}+1-2x}{4} $
$ \Rightarrow 0=\dfrac{16-16x+{{x}^{2}}+1-2x}{4} $
$ \Rightarrow {{x}^{2}}-18x+17=0 $
$ \Rightarrow {{x}^{2}}-17x-x+17=0 $
$ \Rightarrow x\left( x-17 \right)-1\left( x-17 \right)=0 $
$\Rightarrow \left( x-17 \right)\left( x-1 \right)=0 $
So, \[x=17\] or \[x=1\] .
On putting \[x=17\] in equation (3) and equation (5), we get
The length of the base of the given right-angled triangle = \[\left( 17-2 \right)\] cm = 15 cm ……………………………………(7)
The length of the altitude of the given right-angled triangle = \[\left( \dfrac{17-1}{2} \right)\] cm = 8 cm ……………………………………..(8)
Similarly, on putting \[x=1\] in equation (3) and equation (5), we get
The length of the base of the given right-angled triangle = \[\left( 1-2 \right)\] cm = -1 cm, which is not possible.
The length of the altitude of the given right-angled triangle = \[\left( \dfrac{1-1}{2} \right)\] cm = 0 cm, which is not possible.
So, \[x=17\] is possible whereas \[x=1\] is not possible.
Therefore, the length of the hypotenuse, altitude, and the base of the given right-angled triangle is 17 cm, 8 cm, and 15 cm.
So, the correct answer is “Option C”.
Note: For solving this type question one must remember the Pythagoras theorem, that is, \[{{\left( Hypotenuse \right)}^{2}}={{\left( Altitude \right)}^{2}}+{{\left( Base \right)}^{2}}\] . We can also solve this question by using the options given below. Check all the options given and figure out the options that satisfy the Pythagoras theorem. But only figuring out the options satisfying the Pythagoras theorem is not sufficient enough to get the correct answer. For instance, here every option is satisfying the Pythagoras theorem formula. Therefore, also check the options that are satisfying the information provided in the question. For instance, here only option (C) is satisfying the information provided in the question i.e., the length of a hypotenuse of a right-angled triangle exceeds the length of its base by 2 cm and exceeds twice the length of the altitude by 1 cm. Hence, option (C) is the correct option.
