14. Which set of side lengths can be used to prove the statement below is false? "In any given triangle, the square of the longest side is equal to the sum of the squares of the two shorter sides."

a) 4 cm, 5 cm, 7 cm

b) 5 cm. 12 cm 13 cm

c) 3 cm, 4 cm, 5 cm

d) 9 cm, 12 cm, 15 cm

The only set of side lengths that proves that the statement "In any given triangle, the square of the longest side is equal to the sum of the squares of the two shorter sides." is false is;

Option A; 4 cm, 5 cm, 7 cm

To answer this question, let us look at each option;

Option A; 4 cm, 5 cm, 7 cm

Thus; 4² + 5² = 51 and 7² = 49

Since the sum of the square of the two shorter sides is not equal to the square of the longest side, then this proves the statement is false.

Option B; 5 cm. 12 cm 13 cm

Thus; 5² + 12² = 169 and 13² = 169

Since the sum of the square of the two shorter sides is equal to the square of the longest side, then this proves the statement is true.

Option C; 3 cm, 4 cm, 5 cm

Thus; 3² + 4² = 25 and 5² = 25

Since the sum of the square of the two shorter sides is equal to the square of the longest side, then this proves the statement is true.

Option D; 9 cm, 12 cm, 15 cm

Thus; 9² + 12² = 225 and 15² = 225

Since the sum of the square of the two shorter sides is equal to the square of the longest side, then this proves the statement is true.

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a) 4 cm, 5 cm, 7 cm

Step-by-step explanation:

The statement "In any given triangle, the square of the longest side is equal to the sum of the squares of the two shorter sides." is only true for triangle rectangles.

To find a counter example, we can try with the options and see which one works as a counterexample:

a) 4 cm, 5 cm, 7 cm

The longest side is 7cm, then the statement says that:

4^2 + 5^2 = 7^2

16 + 25 = 51

while 7^2 = 49

Then for this particular set of sides, the statement is false.

So we already found the counter example, because:

4^2 + 5^2 ≠ 7^2

1/4 because there is 4 grades.