Question

A right triangle, $△ABC$ has an area of $12c{m}^2$ . Which of the following is the possible length of side $AC$ if $AB$ and $BC$ are the perpendicular sides?

• A. $42.02\text{cm}$
• B. $2.71\text{cm}$
• C. $8.5\text{cm}$
• D. $21.17\text{cm}$

Answer 1

The correct option is C $8.5\text{cm}$

Area of a right triangle =$\frac{1}{2}\times$ Product of the perpendicular sides
Product of the perpendicular sides =$2\times$ Area of a right triangle = $2\times 12=24$

Let’s now find the factors for $24$.
The possible combinations are:
$24=1\times 24$

$24=2\times 12$

$24=4\times 6$

$24=8\times 3$

Since these are the possible factors of $24$, these must be the possible lengths of the perpendicular sides.

Let’s construct triangles taking these measures of $AB$ and $BC$, and measure the length of $AC$. Since in the question, it is given that $AB$ and $BC$ are perpendicular to each other, $\angle ABC={90}^{\circ }$.

Case 1 :$AB=24\text{cm},BC=1\text{cm},\angle ABC={90}^{\circ }$
Here, we know two sides of a triangle and the included angle; hence, we can use SAS criterion.

1.First, draw a line segment $AB$ of length $24\text{cm}$.

2.Next, draw an angle of ${90}^{\circ }$ at vertex $B$.


3.Next, cut a $1\text{cm}$ arc with $B$ as the center to get $C$.


4.Join $AC$.


In this case, $AC\approx 24.02\text{cm}$.

Similarly,
for case 2, $BC=1\text{cm},AB=12\text{cm}$, and $\angle ABC={90}^{\circ }$.


In this case, $AC\approx 12.17\text{cm}$

Case 3: $BC=4\text{cm},AB=6\text{cm},\angle ABC={90}^{\circ }$


In this case, $AC\approx 7.21\text{cm}$

Case 4: $BC=8\text{cm},AB=3\text{cm},\angle ABC={90}^{\circ }$

In this case, $AC\approx 8.5\text{cm}$ .
Since option $B$ is the only option, that matches with one of the above four cases, $AC\approx 8.5\text{cm}$ .