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**Exercise 10.1**

**1. Draw a quadrilateral in the Cartesian plane, whose vertices are **** and **** Also, find its area.**

**Ans. **Area of quadrilateral ABCD = Area of + Area of trapezium ABED

=

=

= 26 + 34.8 = 60.8 sq. units

**2. The base of an equilateral triangle with side **** lies along the ****axis such that the mid-point of the base is at origin. Find the vertices of the triangle.**

**Ans.** Given: Length of side of equilateral triangle = . The base of triangle lies along axis and the mid-point of base is at origin so that the coordinates of vertices are and

Let vertices of the third vertex be

The vertices of triangle are , and

**3. Find the distance between P**** and Q**** when (i) PQ is parallel to the ****axis (ii) PQ is parallel to the ****axis.**

**Ans. **Given: P and Q are two points.

PQ =

**(i) **PQ is parallel to axis, then

PQ =

**(ii) **PQ is parallel to axis, then

PQ =

**4. Find the point on the ****axis, which is equidistant from the points (7, 6) and (3, 4).**

**Ans. **Let P be any point on the axis which is equidistant from Q (7, 6) and R (3, 4).

PQ =

=

=

And PR =

=

=

According to question, PQ = PR

=

Squaring both sides, =

Therefore, the coordinates of the point are

**5. Find the slope of a line, which passes through the origin and the mid-point of the line segment of joining the points P**** and B (8, 0).**

**Ans. **Here, mid-point of the line segment joining P and Q (8, 0) is

Since the line passes through points (0, 0) and .

Therefore, Slope of the line =

**6. Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and **** are the vertices of a right angled triangle.**

**Ans. **Let A (4, 4), B (3, 5) and C be three vertices of a

Slope of AB =

Slope of BC =

Slope of AC =

Now, Slope of AB x Slope of AC =

This shows that AB AC. Thus is right angled at point A.

**7. Find the slope of the line, which makes an angle of **** with the positive direction of ****axis measured anticlockwise.**

**Ans. **The line makes an angle of with the positive direction of axis.

Now the line makes an angle of with the positive direction of axis.

Slope of the line =

**8. Find the value of **** for which the points **** and **** are collinear.**

**Ans. **Let A B (2, 1) and C (4, 5) be three collinear points.

Slope of AB =

Slope of BC =

According to question, Slope of AB = Slope of BC

**9. Without using distance formula, show that the points **** and **** are the vertices of a parallelogram.**

**Ans. **Let A B (4, 0), C (3, 3) and D be vertices of a quadrilateral ABCD.

Slope of AB = Slope of BC =

Slope of DC = Slope of AD =

Here Slope of AB = Slope of DC

AB DC

And Slope of BC = Slope of AD

BC AD

Therefore, ABCD is a parallelogram.

**10. Find the angle between the ****axis and the line joining the points **** and **

**Ans. **Let A and B be two points. Let Q be the angle which AB makes with positive direction of axis.

Slope of AB =

Also Slope of AB =

**11. The slope of a line is double of the slope of the another line. If tangent of the angle between them is **** find the slopes of the lines.**

**Ans. **Given: . Let the slopes of two lines be and

Taking

and

Taking

and

Therefore, the slopes of lines are and or 1 and

**12. A line passes through **** and **** If slope of the line is **** show that **

**Ans. **Let A and B be two points. It is given that Slope of AB =

Slope of AB = = (given)

**13. Is three points **** and **** lies on a line, show that **

**Ans. **Let A B and C be three points lie on the line.

Slope of AB = Slope of BC =

Slope of AB = Slope of BC (given)

**14. Consider the following population and year graph, find the slope of the line AB and using it, find what will be the population in the year 2010?**

**Ans.** Given: The points on the line are A (1985, 92) and B (1995, 97).

Slope of AB =

Let the population in year 2010 be crores. Then C lies on the line.

Slope of BC =

Since points A, B and C lie on the line.

Slope of AB = Slope of BC

Therefore, population in 2010 will be 104.5 crores.

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