‘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 \(2 : 3\).

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

Coordinate Geometry

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

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

Class 10 Mathematics · Chapter 7

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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

Try one from this chapter

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

Flashcards & interactive practice

Coordinate Geometry

CBSE Class 10 Board Examination community

Summary

Key terms

Extra questions & answers

EXERCISE 7.1

EXERCISE 7.2

Practice quiz · Coordinate Geometry

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