Question

In triangle ABC, angle B is 90 degree. If \[AB=\left( x-3 \right)cm,BC=\left( x+4 \right)cm\] and \[AC=\left( x+6 \right)cm\] find AB.

Answer 1

Hint: We will use the Pythagoras theorem which states that the square of length of hypotenuse is equal to the sum of squares of length of the other two sides. In mathematical form, it is written as \[A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\] . Using this we will get two values of x, and we will get the length of AB by putting x as \[\left( x-3 \right)\] .

Complete step-by-step answer:
Here, it is given that angle B is 90 degree i.e. triangle ABC is right angle triangle. Figure is as shown below:
  

So, now we will be using Pythagoras theorem which is given as \[A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\] i.e. square of length of hypotenuse is equal to sum of squares of length of other two sides.
So, now substituting the values, in the theorem we can write it as
\[{{\left( x+6 \right)}^{2}}={{\left( x-3 \right)}^{2}}+{{\left( x+4 \right)}^{2}}\]
Now, we will use the formula \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\] . We get as
\[{{x}^{2}}+2\cdot x\cdot 6+{{\left( 6 \right)}^{2}}={{x}^{2}}+2\cdot x\cdot \left( -3 \right)+{{\left( -3 \right)}^{2}}+{{x}^{2}}+2\cdot x\cdot 4+{{\left( 4 \right)}^{2}}\]
On further simplification, we get as
\[{{x}^{2}}+12x+36={{x}^{2}}-6x+9+{{x}^{2}}+8x+16\]
Taking all the terms on left hand side, we get as
\[{{x}^{2}}+12x+36-{{x}^{2}}+6x-9-{{x}^{2}}-8x-16=0\]
On solving, we get as
\[-{{x}^{2}}+10x+11=0\]
We will multiply the equation with minus side on both sides, we get as
\[{{x}^{2}}-10x-11=0\]
Now, to find roots of this quadratic equation, we will split the middle terms in such a way that on multiplying we get as \[-11\] and on adding we get \[-10\] . So, we get as
\[{{x}^{2}}-11x+x-11=0\]
Taking x common from first 2 terms and 1 common from last 2 terms. We get as
\[x\left( x-11 \right)+1\left( x-11 \right)=0\]
\[\left( x+1 \right)\left( x-11 \right)=0\]
Thus, we get x as 11 and \[-1\] .
Now, if we take x as \[-1\] , then Length of AB as \[\left( -1-3 \right)=-4\] . We know that length cannot be negative. So, this is not the answer.
Now, if we take x as 11, we get AB as \[\left( 11-3 \right)=8cm\] . Thus, this length is positive so the answer will be 8 cm.
Thus, the length of AB is 8cm.


Note: Be careful while solving the quadratic equation. There are chances of making mistakes in splitting the middle term which leads to getting wrong factors and at last length of AB will be also wrong. So, do not make mistakes. Also, figure is important in this type of problem.