Question

you can use the pythagorean theorem to find the distance between two points on the coordinate plane. lets try it! find the distance between points s and t. first, draw a right triangle with a hypotenuse that connects s and t. next, find the length of each leg. to find the length of the horizontal leg, find the absolute value of the difference of the x - coordinates of the endpoints on that leg: |-3 - 1| = |-4| = 4. to find the length of the vertical leg, find the absolute value of the difference of the y - coordinates of the endpoints on that leg: |4 - 1| = |3| = 3. you can check the lengths you got above by counting the horizontal and vertical distances on the coordinate plane. finally, use the pythagorean theorem, a² + b² = c², to solve for the length of the hypotenuse. let a = 4 and b = 3. a² + b² = c² 4² + 3² = c² 16 + 9 = c² 25 = c² √25 = √c² 5 = c the length of the hypotenuse is the distance between points s and t. so, the distance between the points is 5 units. try it yourself! use the pythagorean theorem to find the distance between each pair of points.

Explanation:

Step1: Assume coordinates of points C and D

Let's assume point C has coordinates $(4, 5)$ and point D has coordinates $(0,-6)$.

Step2: Find length of horizontal leg

Find the absolute - value of the difference of x - coordinates: $|4 - 0|=4$.

Step3: Find length of vertical leg

Find the absolute - value of the difference of y - coordinates: $|5-(-6)|=|5 + 6| = 11$.

Step4: Apply Pythagorean theorem

Use $a^{2}+b^{2}=c^{2}$, where $a = 4$ and $b = 11$. Then $4^{2}+11^{2}=c^{2}$, $16 + 121=c^{2}$, $137=c^{2}$, $c=\sqrt{137}$.

For points V and W, assume V has coordinates $(-2,5)$ and W has coordinates $(3,-3)$.

Step5: Find length of horizontal leg for V and W

Find the absolute - value of the difference of x - coordinates: $|-2 - 3|=| - 5|=5$.

Step6: Find length of vertical leg for V and W

Find the absolute - value of the difference of y - coordinates: $|5-(-3)|=|5 + 3| = 8$.

Step7: Apply Pythagorean theorem for V and W

Use $a^{2}+b^{2}=c^{2}$, where $a = 5$ and $b = 8$. Then $5^{2}+8^{2}=c^{2}$, $25+64=c^{2}$, $89=c^{2}$, $c=\sqrt{89}$.

Answer:

$\sqrt{137}$
$\sqrt{89}$