Question

the length of two sides of right triangle containing the right angle differ by 2cm if the area of the triangle is 24cm², find its perimeter​

Answer 1

Answer:

The perimeter of the triangle is 24 cm

Step-by-step explanation:

Given :

• The length of two sides of right triangle containing the right angle differ by 2cm
• The area of the triangle is 24cm²

To find :

the perimeter of the triangle

Solution :

Let the base of the triangle be 'a' cm

Since the length of two sides of right triangle containing the right angle differ by 2cm,

let the height of the right angled triangle be (a+2) cm

Area of the right angled triangle = ½ × base × height

24 cm² = ½ × a × (a+2)

24 × 2 = a(a + 2)

48 = a² + 2a

a² + 2a – 48 = 0

a² + 8a – 6a – 48 = 0

a(a + 8) – 6(a + 8) = 0

(a + 8) (a – 6) = 0

a = –8, +6

The length can not be negative.

So, base of the triangle = 6 cm

The height of the triangle = 6+2 = 8 cm

By Pythagoras theorem,

hypotenuse² = base² + height²

hypotenuse² = 6² + 8²

hypotenuse² = 36 + 64

hypotenuse² = 100

hypotenuse = √100

hypotenuse = 10 cm

Perimeter :

The perimeter of the triangle is equal to the sum of the lengths of all the three sides.

= 6 cm + 8 cm + 10 cm

= 24 cm

Answer 2

GIVEN

Let x cm be the one of the sides, then (x−2)cm be another side.

Area of triangle =

${24cm}^{2} \: \: given$

Now

$we \: \: know \: \: aera \: of \: a \: triangle = \frac{1}{2} (base \: \times height)$

$= > 24 = \frac{1}{2} \times x \times x(x - 2) \\ = > 48 = {x}^{2} - 2x \\ or {x}^{2} - 2x - 48 = 0$

Solving for equation,we have

$(x + 6)(x - 8) = 0 \\ x = - 6 \: \: or \: \: x = 8$

Since length measure cannot be negative, so

$x = - 6$

$one \: side \: \: = 8cm \\ another \: side \:= x - 2 = 8 - 2 = 6cm$

$apply \: \: pythagoras \: \: theorem \\ {hypotenuse}^{2} = {base}^{2} + {perpendicular}^{2} \\ {hypotenuse}^{2} = \sqrt{( {8}^{2} + {6}^{2} ) } \\ hypotenuse = \sqrt{100} = 10$

there fore, perimeter of triangle

$= su m \: of \: \: all \: the \: \: sides = (6 + 8 + 10)cm = 24cm$