To demonstrate the converse of the Pythagorean theorem, we need to show that three given side lengths can form a right triangle if and only if the square of one side is equal to the sum of the squares of the other two sides.
Let's examine the side lengths provided: [tex]\( 5 \, \text{cm}, 9 \, \text{cm}, 12 \, \text{cm}, \)[/tex] and [tex]\( 15 \, \text{cm} \)[/tex].
We'll consider different combinations of these side lengths to check which one satisfies the Pythagorean theorem ([tex]\(a^2 + b^2 = c^2\)[/tex]).
1. Check if [tex]\(5 \, \text{cm}\)[/tex] and [tex]\(9 \, \text{cm}\)[/tex] can form a right triangle with [tex]\(12 \, \text{cm}\)[/tex]:
[tex]\[ 5^2 + 9^2 = 25 + 81 = 106 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
So, [tex]\(5^2 + 9^2\)[/tex] is less than [tex]\(12^2\)[/tex]. This does not satisfy the Pythagorean theorem.
2. Check if [tex]\(9 \, \text{cm}\)[/tex] and [tex]\(12 \, \text{cm}\)[/tex] can form a right triangle with [tex]\(15 \, \text{cm}\)[/tex]:
[tex]\[ 9^2 + 12^2 = 81 + 144 = 225 \][/tex]
[tex]\[ 15^2 = 225 \][/tex]
So, [tex]\(9^2 + 12^2 = 15^2\)[/tex]. This satisfies the Pythagorean theorem.
Since we have found that [tex]\(9^2 + 12^2 = 15^2\)[/tex], we can conclude that drawing the 9 cm side and the 12 cm side with a right angle between them will result in the 15 cm side fitting to form a right triangle. This satisfies the converse of the Pythagorean theorem.
Therefore, the correct option is:
C. Knowing that [tex]\(9^2 + 12^2 = 15^2\)[/tex], draw the 9 cm side and the 12 cm side with a right angle between them. The 15 cm side will fit to form a right triangle.
