Question

1)
$5^2 + 7^2 = c^2$
$25 + 49 = 74$
$sqrt{74}$
2)
3)
$4^2 + 2^2 = c^2$
4)
$9^2 + 5^2 = c^2$
$sqrt{106}$

Explanation:

Step1: Identify horizontal/vertical change (Q1)

The horizontal change $\Delta x=5$, vertical change $\Delta y=7$.

Step2: Apply Pythagorean theorem (Q1)

$c=\sqrt{\Delta x^2+\Delta y^2}=\sqrt{5^2+7^2}=\sqrt{25+49}=\sqrt{74}$

Step3: Identify horizontal/vertical change (Q2)

The horizontal change $\Delta x=7$, vertical change $\Delta y=0$.

Step4: Calculate length (Q2)

$c=\sqrt{7^2+0^2}=7$

Step5: Identify horizontal/vertical change (Q3)

The horizontal change $\Delta x=4$, vertical change $\Delta y=2$.

Step6: Apply Pythagorean theorem (Q3)

$c=\sqrt{4^2+2^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$

Step7: Identify horizontal/vertical change (Q4)

The horizontal change $\Delta x=9$, vertical change $\Delta y=5$.

Step8: Apply Pythagorean theorem (Q4)

$c=\sqrt{9^2+5^2}=\sqrt{81+25}=\sqrt{106}$

Answer:

1. $\sqrt{74}$
2. $7$
3. $2\sqrt{5}$
4. $\sqrt{106}$