Question

# The perimeter of △ 𝐻𝐺𝐹 must be between what two integers? Explain your reasoning

## The correct answer is: Hence, perimeter is between 4 and 24.

### Answer:

- Hints:
- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c

b < a + c

c < a + b

- while finding possible lengths of third side use below formula

difference of two side < third side < sum of two sides

- Step-by-step explanation:

- Given:

GJ = 3 units

JH = 4 units

FJ = 5 units

- Step 1:
- Find length of third side in GJH.
- In △GJH,

GJ = 3 and JH = 4

According to triangle inequality theorem,

difference of two side < third side < sum of two sides

b – a < c < a + b

4 – 3 < c < 4 + 3

1 < c < 7

Hence, 1 < GH < 7.

Step 2:

- Find length of third side in △FJH.
- In △FJH,

FJ = 5 and GJ = 3

According to triangle inequality theorem,

difference of two side < third side < sum of two sides

b – a < c < a + b

5 – 3 < c < 5 + 3

2 < c < 8

Hence, 2 < FH < 8.

Step 3:

- Find length of third side in △FJG.
- In △FJG,

FJ = 5 and JH = 4

According to triangle inequality theorem,

difference of two side < third side < sum of two sides

b – a < c < a + b

5 – 4 < c < 5 + 4

1 < c < 9

Hence, 1 < FG < 9.

Step 3

Find perimeter of △HGF

Perimeter = HF + JH + GF

Hence,

1 + 1 + 2 < HF + JH + GF < 7 + 8 + 9

4 < HF + JH + GF < 24

- Final Answer:

Hence, perimeter is between 4 and 24.

- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c

- while finding possible lengths of third side use below formula

- Given:

- Step 1:
- Find length of third side in GJH.
- In △GJH,

According to triangle inequality theorem,

difference of two side < third side < sum of two sides

Hence, 1 < GH < 7.

Step 2:

Hence, 2 < FH < 8.

Step 3:

Hence, 1 < FG < 9.

Step 3

Find perimeter of △HGF

Perimeter = HF + JH + GF

Hence,

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