My Qus to ChatGPT:
Is it possible to have a triangle with sides 3 cm, 6 cm and 7 cm? Show steps of working…

ChatGPT response:
No, it is not possible to have a triangle with sides measuring 3 cm, 6 cm, and 7 cm. This is because the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, the sum of the lengths of the sides 3 cm and 6 cm is 9 cm, which is less than the length of the third side, 7 cm. Therefore, it is not possible to form a triangle with these side lengths.

If you have noticed, ChatGPT failed to recognise 9cm is grater than 7cm.

My question to ChatGPT Plus with GPT-4 and code interpreter (advanced data analysis)

Is it possible to have a triangle with sides 3 cm, 6 cm and 7 cm? Explain your logic, and then calculate all inner angles of the triangle.

Response:

To determine if a set of three lengths can form a triangle, we use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For the given lengths, we need to check if this condition holds true for all three possible pairs of sides:

(3 \text{ cm} + 6 \text{ cm} > 7 \text{ cm})

(3 \text{ cm} + 7 \text{ cm} > 6 \text{ cm})

(6 \text{ cm} + 7 \text{ cm} > 3 \text{ cm})

If all these conditions are met, the lengths can form a triangle.

Checking the Conditions

(3 \text{ cm} + 6 \text{ cm} = 9 \text{ cm} > 7 \text{ cm}), so the first condition is met.

(3 \text{ cm} + 7 \text{ cm} = 10 \text{ cm} > 6 \text{ cm}), so the second condition is met.

(6 \text{ cm} + 7 \text{ cm} = 13 \text{ cm} > 3 \text{ cm}), so the third condition is met.

Since all three conditions are met, it is indeed possible to have a triangle with sides 3 cm, 6 cm, and 7 cm.

Calculating the Inner Angles

To calculate the inner angles of the triangle, we can use the law of cosines, which is given by:

[ c^2 = a^2 + b^2 - 2ab \cos(C) ]

where (a), (b), and (c) are the lengths of the sides of the triangle, and (C) is the angle opposite the side of length (c). We can solve for (\cos(C)) and then use the inverse cosine function to find the angle (C). We can do this for each angle of the triangle.

Let’s calculate the inner angles of the given triangle using this method.

(python code)

The inner angles of the triangle with sides 3 cm, 6 cm, and 7 cm are approximately:

Angle opposite the 3 cm side: (25.21^\circ)

Angle opposite the 6 cm side:(58.41^\circ)

Angle opposite the 7 cm side: (96.38^\circ)

These angles satisfy the conditions of a triangle, where the sum of the angles equals (180^\circ), further confirming that a triangle with these side lengths is indeed possible.