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Toolbox · Points & Coordinate Plane

Points & Coordinate Plane

Plot points to get distances to the origin, the axes and between points, plus midpoints and animated translations.

Formula

\begin{aligned}&d_{12}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[2pt]&d_O=\sqrt{x^2+y^2},\quad d_x=|y|,\quad d_y=|x|\\[2pt]&M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)\\[2pt]&(x,\ y)\xrightarrow{\ \pm a\ }(x\pm a,\ y)\\[2pt]&(x,\ y)\xrightarrow{\ \pm b\ }(x,\ y\pm b)\end{aligned}

pH = −lg c(H⁺), and at 25 °C pH + pOH = 14.

Unit lengthpx

Wheel to zoom · drag to pan · double-click to add a point · drag a point to move it

Scroll to zoom, drag empty space to pan, drag a point to move it, double-click to add a point; the unit length is adjustable.

Points

2 / 12

A(-3, 2)Quadrant II
B(2, -1)Quadrant IV

Presets

Calculate

→

Click a row to highlight the matching segment; for the last row pick the endpoints below.

To origin3.606

To x-axis2

To y-axis3

Between two points5.831

PositionQuadrant II

A(-3, 2)

Steps

5 / 5

1Write coordinates
$P(-3,\ 2)$
Read x first, then y, and write them as the ordered pair (x, y).
2Distances to axes
$d_x=|y|=|2|=2,\quad d_y=|x|=|-3|=3$
The distance to the x-axis is the absolute value of y; the distance to the y-axis is the absolute value of x.
3Distance formula
$d_O=\sqrt{x^2+y^2}=\sqrt{(-3)^2+(2)^2}=\sqrt{13}=3.606$
Take ∣x∣ and ∣y∣ as the legs of a right triangle - the hypotenuse is the distance to the origin.
4Write coordinates
$A(-3,\ 2),\quad B(2,\ -1)$
Read x first, then y, and write them as the ordered pair (x, y).
5Distance formula
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(2-(-3))^2+(-1-(2))^2}=\sqrt{34}=5.831$
The differences in x and in y are the two legs; the hypotenuse is the distance between the points.

Concept

pH is the negative common logarithm of c(H⁺); one unit means a tenfold change in concentration. It is meaningful only for dilute solutions.

A strong electrolyte ionises completely so c(H⁺) follows from concentration; a weak one ionises only slightly, so its equilibrium must be solved. That is the fundamental difference.

The equivalence point is not necessarily pH 7: it is 7 for strong–strong, above 7 when a strong base titrates a weak acid (the salt hydrolyses), and below 7 for the reverse. Choose the indicator by the equivalence pH.

Going further

The distance formula is Pythagoras: Join the two points, then draw the horizontal and vertical legs: the hypotenuse of that right triangle is exactly the distance.
Why the absolute value: Distance is a length, so it can never be negative. A point may have a negative y, but its distance to the x-axis is still ∣y∣.
Translation as a vector: Moving 3 units right adds the vector (3, 0); moving 2 units up adds (0, 2). Translation is just adding a vector to the coordinates.
Euclidean vs Manhattan distance: A taxi follows the streets, so its route is ∣x∣+∣y∣ (Manhattan); only a straight flight gives √(x²+y²). They agree only on the axes.

Common mistakes

Decide first whether the electrolyte is strong or weak — that picks the formula.
For mixtures, do not reach for a pH formula; compare amounts first.
Choose the indicator by the equivalence pH, not by the starting pH.
Midpoint coordinates must be divided by 2 - simply adding the two points does not give the midpoint.

Page keywordspoints and coordinate planedistance formulamidpoint formulatranslation of a pointdistance to axesdistance from a point to the x-axisdistance from a point to the y-axisdistance between two points formulahow to find the midpointtranslate a point a units rightpoint translated b units uponline coordinate plane plotterchange unit length on a coordinate plane

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