# NCERT Class 10 Chapter 4 Coordinate Geometry Official CBSE Board Sample Problems Short Answer

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## Short Answer

## Question

**If the point P(x, y) is equidistant from the points** **and** **Prove that** **.**

### Solution

Given,

Applying distance formula

Hence proved

## Question

**Prove that the points (2, -2), (-2, 1) and (5, 2) are the vertices of a right angled triangle. Also find the area of this triangle.**

### Solution

Let and be the given points. So,

Using Distance formula

, so is a right angled triangle in which BC is hypotenuse.

sq. Units

## Question

**Find the ratio in which the point (-3, k) divides the line-segment joining the points (-5, -4) and (-2, 3). Also find the value of k.**

### Solution

Let P divides AB in k: 1.

Then (Using section formula, ]

Hence the required ratio is 2: 1

## Question

**Prove that the points (30), (6, 4) and (-1, 3) are the vertices of a right angled isosceles triangle.**

### Solution

Let the triangle be ABC as shown in figure. Distances are:

Using distance formula.

Here, is isosceles triangle

Consider,

and,

Here,

is a right angled triangle.

: In right & using Pythagoras theorem

Where H = hypotenuse. B = base, P = perpendiculars

## Question

**Let P and Q be the points of trisection of the line segment joining the points** **and** **such that P is nearer to A. Find the coordinates of P and Q.**

### Solution

Let be given points. Let are point of trisection.

P divides AB in the ratio l :2

Coordinates of P are

Q is midpoint of PB. So using midpoint formula coordinates of are or

## Question

**If the point** **divides internally the line-segment joining the points** **and** **in the ratio 3: 4, find the value of** **.**

### Solution

Using section formula,

Similarly, 2

Ilcncc,

## Question

**In figure ABC is a triangle coordinates of whose vertex A are** **D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of MBC and DEF.**

### Solution

Let coordinates of B are . Then using midpoint formula we

Coordinates of B are (2, l) 2

Let coordinates of C are (p, q)

Similarly coordinates of C we have

Coordinates of C are (0, 3)

Area of

sq. units

Coordinates of F are i.e. (1, 2)( Using mid-point formula we

Area of =

sq. units (Area cannot he negative)

## Question

**Find the coordinates of the points which divide the line segment joining** **and** **into four equal parts.**

### Solution

Let points P, Q and R divide AB in to 4 equal parts.

## Question

**In what ratio does the y-axis divide the line segment joining the points** **and****?**

**Also, find the coordinates of the point of intersection.**

### Solution

----------------------

Hence y axis divides PQ in the ratio 4: 3

Coordinates of point Pare

## Question

**Prove that the points (a, 0), (0, b) and (1, 1) are collinear if,**

Solution

If the points are collinear, area of triangle form by these three points is equal to zero.

Area of the triangle

Divide both side by ab

Hence

## Question

**If the points** **and** **form a parallelogram, find the values of x and y.**

### Solution

Let the vertices of a parallelogram taken in order be and. Since diagonals of a parallelogram bisect each other,

Midpoint of AC= Midpoint of BD

and

## Question

**Find a relation between x and y such that the point** **is equidistant from the points** **and**

### Solution

Let be equidistant from the points and

Given,

By distance formula,

## Question

**The figure shows layout of a park. In the region TQRS roses are planted which needs to be fenced. Find the length of wire required to fence the rose bed?**

### Solution

Similarly,

## Question

**In what ratio does the line** **divides the join of** **and** **?**

### Solution

Let divide the join of P and Q in the ratio k: 1

## Question

**Prove that the points** **and** **are collinear.**

### Solution

Area =