Question

the perimeter of a triangular garden is 18 m. if its area is under root 135 m^2 and one of the tree sides is 8 m, find the remaining two sides

Answer 1

Given :-

• Perimeter of ∆ = 18m.
• Area of ∆ = √135m².
• One side of ∆ = 8m.

To Find :-

• The remaining two sides of ∆ ?

Formula to be Remember for ∆ :-

Triangle :--- is a plane figure with three straight sides and three angles.

→ Area of ∆ = 1/2 * Base * Height = 1/2* ab* sinC = 1/2 * bc *sinA = 1/2 * ca* sinB = √( s(s-a)(s-b)(s-c) ) [ where s = (a+b+c)/2 ]

→ There are three special names given to triangles that tell how many sides (or angles) are equal:---

1) Equilateral Triangle :-- Have Three equal sides and Three equal angles, always 60°..

2) Isosceles Triangle :-- Have Two equal sides and Two equal angles..

3) Scalene Triangle :-- No equal sides and No equal angles...

→ Triangles can also have names that tell you what type of angle is inside: ---

1) Acute Triangle = All angles are less than 90°..

2) Right Triangle = Has a right angle (90°)..

3) Obtuse Triangle = Has an angle more than 90°..

→ The three interior angles always add to 180°...

______________________

Solution :-

we will use Heron's formula Here to compare The Area of ∆.

→ s = (Perimeter /2) = (18/2) = 9m.

Putting values now we get :-

→ √( s(s-a)(s-b)(s-c) ) = √135

→ √[9*(9-8)(9-b)(9-c)] = √135

Squaring Both sides we get,

→ 9*1 * (9-b)(9-c) = (√135)² = 135

Dividing both sides by 9, we get,

→ (9-b)(9-c) = 15

→ 81 - 9c - 9b + bc = 15

→ 81 - 9(b + c) + bc = 15

→ 9(b + c) - bc = 81 - 15

→ 9(b + c) - bc = 66 -------- Equation (1).

____________

Also, if Rest two sides are b & c, we get,

→ 8 + b + c = 18

→ (b + c) = 18 - 8 = 10 cm. ------ Equation (2).

____________

Putting value of Equation (2) in Equation (1) , we get,

→ 9 * 10 - bc = 66

→ bc = 90 - 66

→ bc = 24 ------------ Equation (3).

____________

Now, when we Do Factors of Equation (3), we just have to check now, which satisfy Equation (2) also . (sum of Factors will be 10).

So,

→ 24 = ( 1, 24) , (24, 1), ( 2,12), (12,2) , (3,8) ,(8,3) , (4,6),(6,4)

Now, As we can see (4,6) or (6,4) Satisfy the Equation (2).

Hence, we can Conclude That Rest Two sides of ∆ are 4cm & 6cm respectively .

(Nice Question.)

Answer 2

______________________________

$\huge\tt{GIVEN:}$

• The perimeter of a triangular garden is 18 m.
• Its area is under root 135 m^2
• One of the three sides is 8 m

______________________________

$\huge\tt{TO~FIND:}$

• The remaining two sides

______________________________

$\huge\tt{SOLUTION:}$

We know,

↪S = (perimeter/2) = 18/2 = 9 cm

Now,

Putting Heron's formula here..

↪√[s(s-a)(s-b)(s-c)] = √135

↪√[9(9-8)(9-b)(9-c)] = √135

↪[9×1(9-b)(9-c)] = (√135²)

↪135

_________________________

↪ (9-b)(9-c) = 15

↪ 81-9c - 9b + bc = 15

↪ 81 - 9(b+c) + bc = 15

↪ 9(b+c)- bc = 81-15

↪9(b+c)-bc = 66 ___(EQ.1)

we can also say that,

↪8+b+c = 18

↪(b+c) = 10 cm ______(EQ.2)

__________________________

By comparing both equations,

↪9×10 - bc = 66

↪90 - bc = 66

↪bc = 90 - 66

↪bc = 24

__________________________

Factors of 24 might be (6,4)

24 = 6 & 4 cm

Hence,

The other two sides are 4 cm & 6 cm

__________________________