Question

Can you explain to me what is the answer is

Answer 1

We can calculate the distance between two points in the plane, like H, I and J, with the following formula:

`D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}`

This is a derivation of the Pythagorean theorem and tells us that the square of the distance is equal to the sum of the squares of the difference between the coordinates of each point.

When the distance is only in one dimension (for example, the distance between I and J), we can simplify the formula to:

`D=\sqrt[]{(x_2-x_1)^2}=|x_2-x_1|`

For example, if we calculate the distance between H and I with the general formula, we get:

`\begin{gathered} D=\sqrt[]{(x_h-x_i)^2+(y_h-y_i)^2_{}} \\ D=\sqrt[]{(2-2)^2+(5-9)^2} \\ D=\sqrt[]{0^2+(-4)^2} \\ D=\sqrt[]{0+16} \\ D=\sqrt[]{16} \\ D=4 \end{gathered}`

We could have just calculated the same as:

`D=|y_h-y_i|=|5-9|=|-4|=4`

Now, we already know the distance between H and I.

We will calculate the distance between I and J with the simplified formula, as they have the same y-coordinate:

`D=|x_j-x_i|=|7-2|=|5|=5`

Now, we have to use the general formula to calculate the distance between J and H (NOTE: we can also use the distances already calculated and apply the Pythagorean theorem).

`\begin{gathered} D=\sqrt[]{(x_j-x_h)^2+(y_j-y_h)^2} \\ D=\sqrt[]{(7-2)^2+(9-5)^2} \\ D=\sqrt[]{5^2+4^2} \\ D=\sqrt[]{25+16} \\ D=\sqrt[]{41} \\ D\approx6.4 \end{gathered}`

Then we have:

Distance between I and J = 5 units

Distance between H and I = 4 units

Distance between H and J = 6.4 approximately.

Answer: The correct option is Option C, as it states the distances for the segments correctly.