‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

Find the midpoint of the segment joining \((2, 3)\) and \((4, 7)\).

Midpoint \(= \left(\dfrac{2+4}{2},\ \dfrac{3+7}{2}\right) = (3, 5)\).

Write the distance formula between points \((x_1, y_1)\) and \((x_2, y_2)\).

Distance \(= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Find the distance between the points \((3, 4)\) and \((0, 0)\).

\(\sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) units.

Write the section formula for a point dividing a segment in ratio \(m : n\).

The point is \(\left(\dfrac{mx_2 + nx_1}{m + n},\ \dfrac{my_2 + ny_1}{m + n}\right)\).

Write the midpoint formula.

The midpoint of the segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)\).

Find the midpoint of the segment joining \((2, 3)\) and \((4, 7)\).

\(\left(\dfrac{2 + 4}{2},\ \dfrac{3 + 7}{2}\right) = (3, 5)\).

How can you check whether three points are collinear using coordinates?

They are collinear if the sum of the distances of two pairs equals the third, or if the area of the triangle they form is zero.

In what ratio does the point \((4, 0)\) divide the segment joining \((1, 0)\) and \((7, 0)\)?

Using the section formula on x-coordinates: \(4 = \dfrac{7m + 1n}{m + n}\) gives \(m : n = 1 : 1\), so it is the midpoint.

Find the coordinates of the point dividing the join of \((-1, 7)\) and \((4, -3)\) in the ratio \(2 : 3\).

\(\left(\dfrac{2\cdot4 + 3\cdot(-1)}{5},\ \dfrac{2\cdot(-3) + 3\cdot7}{5}\right) = \left(\dfrac{8 - 3}{5},\ \dfrac{-6 + 21}{5}\right) = (1, 3)\).

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Chapter 7: Coordinate GeometryClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

Q: Write the distance formula between points \((x_1, y_1)\) and \((x_2, y_2)\).

\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Try one from this chapter

The distance between two points (x₁, y₁) and (x₂, y₂) is:

Summary

Coordinate geometry connects algebra with geometry by locating points on a plane using ordered pairs \((x, y)\). The chapter introduces the distance formula to find the length between two points, and the section formula to find the coordinates of a point dividing a line segment in a given ratio (including the midpoint as a special case). These tools let students compute lengths, verify shapes and divide segments without drawing, turning geometric questions into straightforward algebraic calculations.

Distance formulaSection formulaMidpoint formula

Key terms

Distance formula

\(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).

Section formula

Coordinates of a point dividing a segment in ratio \(m:n\).

Midpoint

The point dividing a segment in the ratio \(1:1\).

Coordinate plane

A plane with x- and y-axes for locating points.

Extra questions & answers

EXERCISE 7.1

EXERCISE 7.2

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Distance between two points

x₁y₁x₂y₂

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