Question

4. In the figure below, ABC is a right triangle. The length of

AB is 6 units and the length of CB is 3 units. What is the length, in
units, of AC?

Answer 1

`\textbf{The length of $AC$ = \large{$3√(3)$}}`

Explanation:

Use Pythagoras Theorem.

For any right-angled triangle, with sides,
`\textup{$a, b, c$ we have:\\}`

`[tex]\begin{centre}$$ a^2 + b^2 = c^2 $$ \\\end{centre}`[/tex]

where,
`$a , b $` are the length of the sides and
`$c$` is the hypotenuse.

Here,
`$AB$` is the hypotenuse and
`$AC$` and
`$CB$` are its sides. Therefore from Pythagoras Theorem we have:

`$6^2 = 3^2 + AC^2$\\$\implies 36 = 9 + AC^2 $\\$\implies AC^2 = 27$\\$\implies AC = √(27) = 3√(3) \hspace{25mm} \textup{(Eliminating -27 as distance cannot be negative)}`

So, we say the length of the other side
`$AC$` is
`$3√(3)$` units.

Answer 2

Length of the side AC =
`3  √(3)` units

Explanation:

The given triangle ABC is a right triangle.

Here, AB = 6 units ( hypotenuse)

CB = 3 units (base)

Now, as the triangle is a right triangle, so by

PYTHAGORAS THEOREM

In a right triangle,

`(Base)^(2)  + (Perpendicular)^(2)  = (Hypotenuse)^(2)`

So, here in ΔABC:
`(BC)^(2)  + (AC)^(2)  = (AB)^(2)`

or,
`(3)^(2)  + (AC)^(2)  = (6)^(2)`

⇒
`AC = √(36 - 9)  =  √(27) = 3  √(3)`

or, the length of the side AC =
`3  √(3)` units