Question

in the triangle shown, the value of b is greater than or equal to 3 and less than or equal to 5. noor says the value of c could be 4. logan says the value of c could be greater than 5. who is correct? explain.

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, $c^{2}=b^{2}+2^{2}=b^{2}+4$.

Step2: Find the range of $c^{2}$

When $b = 3$, $c^{2}=3^{2}+4=9 + 4=13$, so $c=\sqrt{13}\approx3.61$. When $b = 5$, $c^{2}=5^{2}+4=25 + 4=29$, so $c=\sqrt{29}\approx5.39$.

Step3: Analyze the statements

Since $\sqrt{13}\leq c\leq\sqrt{29}$, $c$ can be greater than $4$ and can also be greater than $5$. So both Noor and Logan are correct.

Answer:

Both Noor and Logan are correct. Noor is correct because $c$ can take values such that it can be equal to $4$ (as $c$ ranges from approximately $3.61$ to $5.39$). Logan is correct because $c$ can be greater than $5$ (since the upper - bound of $c$ is approximately $5.39$).