Fundamental Geometrical Concepts - Exercise 16(B)

Number of lines that can pass through any given point is

1. 0
2. 1
3. 2
4. infinite

Answer

From a single point, a line can be drawn in any direction, so infinitely many lines are possible.

Hence, option 4 is the correct option.

Two planes intersect in

1. a point
2. a line
3. a pair of lines
4. a pair of parallel lines

Answer

When two planes meet, their common part is always a straight line.

Hence, option 2 is the correct option.

Two lines which never meet are called

1. concurrent lines
2. parallel lines
3. intersecting lines
4. rays

Answer

Parallel lines always remain at equal distance and never intersect.

Hence, option 2 is the correct option.

Number of lines that can be drawn passing through two given points in a plane is

1. 0
2. 1
3. 2
4. infinite

Answer

Only one unique straight line can pass through two fixed points.

Hence, option 2 is the correct option.

In how many points do two intersecting lines meet?

1. 0
2. 1
3. 2
4. infinite

Answer

Intersecting lines cross each other at exactly one point.

Hence, option 2 is the correct option.

Fill in the blanks:

(i) A ............... has no dimensions.

(ii) A line segment extended endlessly in one direction is called a ............... .

(iii) The points lying on the same line are said to be ............... .

(iv) The point at which three or more lines lying in a plane, intersect is called ............... .

(v) A smooth flat surface which extends endlessly in all the directions is called a ............... .

Answer

(i) A point has no dimensions.

(ii) A line segment extended endlessly in one direction is called a ray.

(iii) The points lying on the same line are said to be collinear.

(iv) The point at which three or more lines lying in a plane, intersect is called the point of concurrence.

(v) A smooth flat surface which extends endlessly in all the directions is called a plane.

Write true (T) or false (F):

(i) Two points are always collinear.

(ii) A line has no end points.

(iii) A ray extends endlessly.

(iv) Two lines in a plane always intersect.

(v) Only one line can be drawn passing through two given points in a plane.

Answer

(i) True
Reason — Any two points in space can always be connected by a single unique straight line. Therefore, they are always collinear by default.

(ii) True
Reason — A line is defined as extending infinitely in both directions. Since it never stops, it cannot have any endpoints.

(iii) True
Reason — A ray has one starting point but extends infinitely in the other direction.

(iv) False
Reason — Two lines in a plane only intersect if they are not parallel. Parallel lines stay at the same distance apart and will never meet.

(v) True
Reason — While many lines can pass through a single point, once we have two fixed points, they "lock" the direction, allowing only one unique straight line to pass through both.

Vasu is given a diagram which contains various lines named l, m, n, p and q.

(1) Which of the following is not a pair of intersecting lines ?

1. p and q
2. p and n
3. l and n
4. m and l

(2) Which of the following forms a set of concurrent lines ?

1. l, p, n
2. m, q, n
3. l, n, q
4. q, m, p

(3) Which of the following is a pair of parallel lines ?

1. l and m
2. n and p
3. q and l
4. m and q

(4) How many lines can be drawn passing through the point of intersection of the lines q and n?

1. 0
2. 1
3. 2
4. infinite

Answer

(1)

Lines p and n are running side-by-side and maintain the same distance apart. They are parallel lines, so they will never intersect. All other pairs (p and q, l and n, m and l) clearly cross each other at some point.

Hence, option 2 is the correct option.

(2)

Concurrent lines are three or more lines that all pass through the exact same point. In the diagram, lines q, m, and p all meet at a single shared intersection point in the center-right area.

Hence, option 4 is the correct option.

(3)

Lines n and p are drawn such that they will never meet, no matter how far they are extended. Therefore n and p are parallel lines.

Hence, option 2 is the correct option.

(4)

The intersection of q and n is a single point. Based on the fundamental rules of geometry, an infinite number of lines can be drawn through any single point in a plane.

Hence, option 4 is the correct option.

Assertion: In the figure, l is a line and AB is a line segment.

Reason: A line segment is the part of a line between two points.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

Figure l has arrows on both ends, which is the geometric symbol for a line (extending infinitely).

Figure AB has two distinct endpoints, which is the definition of a line segment. So, the Assertion is true.

The statement in Reason is mathematically correct definition of a line segment and it perfectly explains why AB is called a line segment.

Hence, option 1 is the correct option.

Assertion: In the figure, the three lines, l, m and n are concurrent.

Reason: The maximum number of points of intersection of three lines drawn in a plane is 3.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is false but Reason (R) is true.

Explanation

Concurrent lines must all pass through the same single point. In the figure, the lines intersect at three different points (A, B, and C), forming a triangle. Therefore, they are not concurrent.
The Assertion is false.

If three lines are not parallel and not concurrent, they intersect at exactly 3 points (A with B, B with C, and C with A). This is the maximum possible.
The Reason is true.

Hence, option 4 is the correct option.

Consider the figure shown.

Which of these statements is true?

1. TV is a line; TU and UV are line segments
2. TU and UV are lines; TV is a line segment
3. TV and UV are lines; TU is a line segment
4. TV and TU are lines; UV is a line segment

Answer

To answer this question, let us recall the definitions:

A line segment is the straight path between two fixed end points and has a definite length. It is shown without any arrows.

A line is a line segment extended endlessly in both directions. It has no end points and is shown with arrows at both ends.

A ray is a line segment extended endlessly in one direction. It has one end point and is shown with an arrow at one end only.

Now, observing the given figure:

TV has arrows extending in both directions (beyond T and beyond V). So, TV is a line.

TU is drawn between the points T and U without any arrows extending beyond them. So, TU is a line segment.

UV is drawn between the points U and V without any arrows extending beyond them. So, UV is a line segment.

Hence, option 1 is the correct option.

Consider the figure shown.

How many pair(s) of intersecting lines are in the figure?

1. 1
2. 3
3. 4
4. infinitely many pairs

Answer

To answer this question, let us recall the definition:

Two lines having a common point are called intersecting lines. Two lines in a plane which have no common point are called parallel lines.

Now, observing the given figure:

The figure contains 4 lines in total — 2 horizontal lines and 2 vertical lines.

Let us name the two horizontal lines as h₁ and h₂, and the two vertical lines as v₁ and v₂.

The two horizontal lines (h₁ and h₂) are parallel to each other, so they do not intersect.

The two vertical lines (v₁ and v₂) are also parallel to each other, so they do not intersect.

However, each horizontal line intersects each vertical line. The pairs of intersecting lines are :

(i) h₁ and v₁

(ii) h₁ and v₂

(iii) h₂ and v₁

(iv) h₂ and v₂

So, the total number of pairs of intersecting lines = 4.

Hence, option 3 is the correct option.