POINTS ON A PLANE

Rene Descartes 1596-1650

[color=#1155cc][size=150]Descartes is considered to be the father of coordinate geometry , which combines algebra and geometry and brings them both onto the same table for us to concretise our mathematical ideas.[/size][/color]

[size=150][color=#9900ff]Philosopher and mathematician René Descartes is famous for his phrase [i][u]“I think; therefore I am.”[/u][/i][/color][/size]

[color=#9900ff]WHAT IS MATHEMATICS ? -Essay published in The New Yorker - March 21 -Author Alec Wilkinson.[br][br][size=150]"So long as there is a world with a horizontal and a vertical axis, a sky and a horizon, it is inviolable and as true as anything that can be thought."-Alec Wilkinson[/size][/color]

[b]Alec Wilkinson[/b] (born 1952) is a writer who has been on the staff of [i][url=https://en.wikipedia.org/wiki/The_New_Yorker]The New Yorker[/url][/i] since 1980. According to [i][url=https://en.wikipedia.org/wiki/The_Philadelphia_Inquirer]The Philadelphia Inquirer[/url] [/i]he is among the "first rank of" contemporary American (20th and early 21st century) "literary journalists...(reminiscent) of [url=https://en.wikipedia.org/wiki/Naipaul]Naipaul[/url], [url=https://en.wikipedia.org/wiki/Norman_Mailer]Norman Mailer[/url] and [url=https://en.wikipedia.org/wiki/James_Agee]Agee[/url]."- Wikipedia

CARTESIAN PLANE

[size=150][color=#660000]As we know we can identify a point in a plane by its coordinates ,the x and y coordinates.Let us explore different points in the plane. The applet below shows a plane and two points A and B. Notice how the coordinates are found . Play with this applet by dragging the two points wherever you like. On a touch-devices you can keep your finger on the point and move by sliding it. On other devices you can drag using the mouse-click.[br][br]Next move the two points A and B to different quadrants and observe the change in their coordinates.[/color][/size]

QUIZ

[size=150]Let us now answer a few very simple questions.[/size]

Q.1

If A is in third quadrant then

Only y-coordinate is negative Both the coordinates are negative Only x-coordinate is negative Both the coordinates are positive

Q.2

Drag the point A or B so that both the points are equal. Observe what happens to their coordinates.

Both the coordinates match. Only one coordinate of both match

Q.3

If [math]\left(x,y\right)=\left(2x+6,3y+4\right)[/math] then

[math]x=2[/math] and [math]y=-3[/math] [math]x=-3[/math] and [math]y=2[/math] [math]x=-6[/math] and [math]y=-2[/math] [math]x=3[/math] and [math]y=-3[/math]

[size=100][size=150][color=#a64d79]In the applet below put the white point so that it plots the given point.[/color][/size][/size]

Distance Between Two Points

Distance between the points P and Q : [math]PQ=\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}[/math][br][br]

Let us have a little fun.

[color=#0000ff]In the following applet, drag the two points A and B so that the distance between them is exactly equal to 5. Try to make the coordinates all different from each other. Then try to make the distance equal to 10. Observe the values of [math]\left(x_2-x_1\right)[/math] and [math]\left(y_2-y_1\right)[/math][/color]. [color=#0000ff]You will see a pattern.[/color]

Let us have some more practice.

Do the exercise below. Try to do it mentally as far as possible. Repeat the exercise by clicking "Try another question like this one" tab.

Segment Formula

The coordinate of the point dividing the segment joining two points [math]\left(x_1,y_1\right)[/math] and [math]\left(x_2,y_2\right)[/math] in the ratio [math]\frac{m}{n}[/math] is given by [math]\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)[/math]

Fun with mid point.

[color=#1e84cc]In the applet , drag the points A,B,C to different positions and make C as the mid point of the segment AB.[/color]

Some more practice with segment formula.