Question

Q.1)The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.

Q.2)Determine all pairs of consecutive even positive integers, both of which are greater than 5 such that their sum is less than 23.

No spam
Irrelevant answers will be reported on spot​

Answer 1

$\huge\underline{{Questions:}}$

Q.1)The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.

Q.2)Determine all pairs of consecutive even positive integers, both of which are greater than 5 such that their sum is less than 23.

$\huge\underline{{Answers:}}$

1. Let the length of shortest side be x cm.

According to the given information,

• Longest side =2x
• Shortest side =2x cm
• And third side = 2+ Shortest side
• =(2+x) cm

Perimeter of triangle

$= x + 2x + (x + 2) = 4x + 2$

According to the question,

$perimeter > 166cm \\ = 4x + 2 > 166 \\ = 4x > 166 - 2 \\ = 4x > 164 \\ x > \frac{164}{4} = 41cm$

Hence, the minimum length of shortest side be 41cm.

2. Let the required consecutive even integers be x and x + 2.

Then,

$x > 5and \: x + (x + 2) < 23 \\ = x > 5and \: 2x < 21 \\ = 5 < x \: and \: x < 10.5 \\ = 5 < x < 10.5$

Therefore, x can take the even integral value 6, 8 and 10.

Hence, the required pairs of even integers are (6, 8), (8, 10) and (10, 12).

Answer 2

$\huge \bold{ \underline{ \underline{{Question}}}}$

The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.

Let the length of shortest side = ‘x’ cm

According to the question,

The longest side of a triangle is twice the shortest side

⇒ Length of largest side = 2x

Also, the third side is 2 cm longer than the shortest side

⇒ Length of third side = (x + 2) cm

Perimeter of triangle = sum of three sides

= x + 2x + x + 2

= 4x + 2 cm

Now, we know that,

Perimeter is more than 166 cm

⇒ 4x + 2 ≥ 166

⇒ 4x ≥ 164

⇒ x ≥ 41

Hence, minimum length of the shortest side should be = 41 cm.

$\huge \bold{ \underline{ \underline{{Question}}}}$

Determine all pairs of consecutive even positive integers, both of which are greater than 5 such that their sum is less than 23.

Let 2x and 2x +2 are two consecutive even positive integers. Then a/C to question, 2x>5 =>x> 5/2

and 2x + 2 > 5

=> 2x > 5-2 = 3

=> 2x > 3

=> x> 3/2

and, sum of 2x and 2x +2 < 23

2x + 2x +2 < 23

=> 4x < 23 -2

=> 4x < 21

=> x < 21/4

now plotting these all value on numberline,

5/2 < x < 21/4

hence, possible value of x = 3, 4, 5

• So, when x = 3 then, (2x3,2x3+2) = (6,8)
• when x = 4 then, (2x4,2x4+2) = (8,10)
• when x= 4 then, (2x5,2x5+2) = (10,12)

Hence, required pair is (6,8), (8,10) and (10,12)

$\huge\bold{ \underline{ \underline{ \blue{@WildCat7083}}}}$