Question

The hypotenuse and a side of a right angled triangle are 13cm and 5cm. Find

its area​

Answer 1

Answer:

30 cm²

Step-by-step explanation:

Let the triangle be ABC.

In Pythagoras' theorem,

AC² = AB² + BC²

where AC is the hypotenuse and AB is the base.

You know both of them, but not the height.. To find the area of a triangle, the formula is

x base x height

AC² = AB² + BC²

(13)² = (5)² + BC²

169 = 25 + BC²

169 - 25 = BC²

144 = BC²

Now, the square,

BC = √144

BC = ± 12

But because this is a measurement unit, there can be no negative values. Thus, BC = 12 cm.

Now, the area.

Area of ABC =  x base x height

                     =  x AB x BC

                     =  x 5 x 12

                     = 30 cm²

P.S I suggest you draw a triangle and name its sides to help you visualize this more.

Answer 2

Given

The triangle mentioned is a right angled triangle since it has hypotenuse.

Hypotenuse ⇒ 13 cm

Base ⇒ 5 cm

$\rule{300}{1}$

To Find

The are of the given triangle.

$\rule{300}{1}$

Solution

To find the are we must find one measure of this triangle.

To do so we must apply Pythagorean Theorem because this is a right angled triangle.

Pythagorean Theorem ⇒ Base² + Height² = Hypotenuse

Let the height be 'x'

Let's solve this equation to find the height ⇒ 5² + x² = 13²  

Step 1: Find the actual value of 5² and 13²

5² + x² = 13³

25 + x² = 169

Step 2: Subtract 25 from both sides of the equation.

25 + x² - 25 = 169 - 25

x² = 144

Step 3: Find the square root of 144.

x² = 144

x = 12

∴ The value of height is 12 cm.

Now we will find the area of the given triangle using the Heron Law.

Heron Law ⇒ $\sqrt{s(s-a)(s-b)(s-c)}$

Over here 's' is the half of the perimeter. So we must find the perimeter of this triangle in order to obtain the area.

Perimeter of Triangle ⇒ Side A + Side B + Side C

Perimeter of given Triangle ⇒ 5 + 12 + 13 = 30

Half of Perimeter ⇒ 30÷2 = 15

So the vale of 's' in the heron's law in our question would be 15.

$\sqrt{s(s-a)(s-b)(s-c)}$

$\sqrt{15(15-5)(15-12)(15-13)}$

$\sqrt{15(10)(3)(2)}$

$\sqrt{15\times 10\times 3\times 2}$

$\sqrt{900}$

$30$

∴ The area of the given triangle is 30 cm²

$\rule{300}{1}$