Pythagoras and trigonometry

To make sense of the distance formula, imagine two points on the coordinate plane: $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$.

This connects to Coordinate Geometry basics. If you draw the straight segment from $P_1$ to $P_2$, that segment’s length is the distance $d$.

We can treat the segment between the points as the hypotenuse of a right triangle.

1. Start at $P_1$ and draw a line straight across (horizontal).
2. From $P_2$, draw a line straight up/down (vertical).
3. Where those lines meet, call the point $P_3(x_2,y_1)$. That corner is a right angle.

Now we can apply Right triangle geometry to the situation.

The bottom (horizontal) side measures how far apart the $x$-coordinates are.

We write that horizontal distance as an absolute difference: $\text{Base length}=|x_2-x_1|$

The vertical side measures how far apart the $y$-coordinates are.

That vertical distance is: $\text{Height length}=|y_2-y_1|$

In the Pythagorean Theorem, $a^2+b^2=c^2$, the sides $a$ and $b$ are the two legs (the sides that meet to make the 90° angle).

Comparing with our coordinate-plane triangle:

• $|x_2-x_1|$ is one leg (horizontal).
• $|y_2-y_1|$ is the other leg (vertical).
• $d$ is the hypotenuse (the slanted distance between the points).

Check each incorrect idea:

• These expressions give lengths, not coordinate pairs.
• They are not angle measures (which would be in degrees or radians).
• The hypotenuse is the whole distance $d$, not each separate difference.

So, both on the diagram and in the formula, $|x_2-x_1|$ and $|y_2-y_1|$ are the horizontal and vertical distances—meaning the two legs of the right triangle.

We can answer by matching the distance formula to Pythagoras.

Pythagorean form for hypotenuse $c$: $c=\sqrt{a^2+b^2}$

Distance between two points: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Both are “square root of a sum of two squares,” so the parts correspond.

By direct comparison, the squared pieces in the distance formula play the role of $a^2$ and $b^2$:

$\begin{array}{rl}{a}^2& =(x_2-x_1{)}^2\\ b^2& =(y_2-y_1{)}^2\end{array}$

Taking square roots gives the side lengths (as nonnegative values):

$\begin{array}{rl}a& =|x_2-x_1|\\ b& =|y_2-y_1|\end{array}$

In $a^2+b^2=c^2$, the quantities $a$ and $b$ are the two legs that meet at the right angle.

So $|x_2-x_1|$ (horizontal change) and $|y_2-y_1|$ (vertical change) must be the lengths of those two legs in the coordinate-plane triangle.

Now remove the options that conflict with the algebra:

• Option A: Coordinates are written as $(x,y)$, but these are single-number distances.
• Option B: $a$ and $b$ are side lengths in Pythagoras, not angle measures.
• Option D: The hypotenuse corresponds to $d$, not to either absolute difference by itself.

The comparison shows the terms represent the two perpendicular side lengths that form the right angle.

We’re told the distance formula is connected to the Pythagorean Theorem:

$a^2+b^2=c^2$

In $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, the value $d$ plays the role of the hypotenuse $c$. That means the squared expressions inside the root must correspond to $a^2$ and $b^2$ (the other two sides).

In Coordinate Geometry, a coordinate is an ordered pair like $(x,y)$. But $|x_2-x_1|$ and $|y_2-y_1|$ each simplify to one number (a length). Since they are not ordered pairs, they cannot describe coordinates.

Angles describe rotation and are measured in degrees or radians. The expressions $|x_2-x_1|$ and $|y_2-y_1|$ describe how far you move horizontally and vertically between points, so they represent lengths rather than angles.

The hypotenuse is the side opposite the right angle and is the longest side. In the distance formula, that full length is $d$. Since $|x_2-x_1|$ and $|y_2-y_1|$ are the parts used to compute $d$, they cannot be the hypotenuse themselves.

After eliminating the others, the remaining choice is that they are the leg lengths. On a coordinate grid, $|x_2-x_1|$ gives the horizontal leg and $|y_2-y_1|$ gives the vertical leg, matching the two legs in the Pythagorean Theorem.