Question

△ABC is a right triangle with right angle C. Side AC¯¯¯¯¯¯¯¯ is 6 units longer than side BC¯¯¯¯¯¯¯¯. If the hypotenuse has length 217−−√ units, find the length of AC¯¯¯¯¯¯¯¯.

Answer 1

Given:

In △ABC is a right angle triangle.

AC is 6 units longer than side BC.

`Hypotenuse=2√(17)`

To find:

The length of AC.

Solution:

Let the length of BC be x.

So, Length of AC = x+6

According to the Pythagoras theorem, in a right angle triangle

`Hypotenuse^2=Base^2+Perpendicualar^2`

△ABC is a right angle triangle and AC is hypotenuse, so

`(2√(17))^2=(x)^2+(x+6)^2`

`68=x^2+x^2+12x+36`
`[\because (a+b)^2=a^2+2ab+b^2]`

Subtract 68 from both sides.

`0=2x^2+12x+36-68`

`0=2x^2+12x-32`

`0=2(x^2+6x-16)`

Divide both sides by 2.

`x^2+6x-16=0`

Splitting the middle term, we get

`x^2+8x-2x-16=0`

`x(x+8)-2(x+8)=0`

`(x+8)(x-2)=0`

`x=-8,2`

Side cannot be negative, so x=2 only.

Now,

`AC=x+6`

`AC=2+6`

`AC=8`

Therefore, the length of AC is 8 units.