Question

A right angled triangle ABC is constructed with BC = 3 $\mathrm{cm}$ and AB + AC = 9$\mathrm{cm}$. If B is the right angled, then find the length of AC (in cm).

• A. 5

Answer 1

The correct option is A 5
The steps to construct the triangle is as follows.

Step 1: The segment BC is drawn.

Step 2: Construct $\angle CBX={90}^{\circ }.$

Step 3: An arc of radius 9 cm (AB+AC) is drawn from B on BX such that BD = 9 cm.

Step 4: The perpendicular bisector (MN) of CD is constructed which intersects BD at point A.

Step 5: A and C are joined.

$△ABC$ is the required triangle.

In $△ABC$, let the length of AC be $x.$
$AB+AC=9,AB=9-x.$

By using Pythogoras' theorem
$A{B}^2+B{C}^2=A{C}^2.$

$\therefore (9-x{)}^2+3^2=x^2\Rightarrow 9^2-18x+x^2+9=x^2\Rightarrow -18x+x^2-x^2=-9-81\Rightarrow -18x=-90\Rightarrow x=\frac{-90}{-18}\Rightarrow x=5$.

Thus, the length of AC is 5 cm.