Question

If the area of the right triangle $ABC$ is $30$ sq. units and the base is 12 units, then

• A. $AB=5$ units
• B. $AB=6$ units
• C. The perimeter of $△ABC=25$ units
• D. The perimeter of $△ABC=30$ units

Answer 1

Answer: (A), (D)

The perimeter of $△ABC=30$ units

Let, $AB=x$ and $AC=y$
Given, the area of $△ABC=30$ sq. units

Step 1: Let's find $x:$
We know that, the area of a triangle $=\frac{1}{2}\times \text{base}\times \text{height}$
$\Rightarrow$ The area of the $△ABC$
$=\frac{1}{2}\times 12\times x$
$\Rightarrow 30=6\times x$
Divide both sides by $6$
$\Rightarrow \frac{30}{6}=\frac{6\times x}{6}$
$\Rightarrow \frac{6\times 5}{6}=x$
$\Rightarrow 5=x$
$\Rightarrow x=5$

Step 2: Let's find $y:$
Using pythagorean theorem on $△ABC,$ we get
$A{C}^2=A{B}^2+B{C}^2$
$\Rightarrow y^2=5^2+{12}^2$
$\Rightarrow y^2=25+144$
$\Rightarrow y^2=169$
Taking square root both sides
$\Rightarrow \sqrt{y^2}=\sqrt{169}$
$\Rightarrow y=\sqrt{{13}^2}(\because {13}^2=169)$
$\Rightarrow y=13$

Step 3: Find the perimeter:
Perimeter of the $△ABC$
$=AB+BC+CA$
$=5+12+13$
$=30$ units

Hence, options (a.) and (d.) are correct.