Fundamental Geometrical Concepts

A line segment has two end points and a definite length, whereas a ray has one end point (its initial point) and extends endlessly in one direction.

The diagonal path joining B and C is a line segment, while the parts starting from B towards A and from C towards D extend endlessly.

∴ Line segment — $\mathbf{\overline{BC}}$ ; Rays — $\mathbf{\overrightarrow{BA} \text{ and } \overrightarrow{CD}}$.

On the right the points D, E and F are joined to give line segments, while three arms extend endlessly towards the left.

∴ Line segments — $\mathbf{\overline{DE}, \overline{EF} \text{ and } \overline{DF}}$ ; Rays — $\mathbf{\overrightarrow{DG}, \overrightarrow{EH} \text{ and } \overrightarrow{FI}}$.

The point O is joined to N and P giving line segments, while the arms from N towards M and from P towards Q extend endlessly.

∴ Line segments — $\mathbf{\overline{ON} \text{ and } \overline{OP}}$ ; Rays — $\mathbf{\overrightarrow{NM} \text{ and } \overrightarrow{PQ}}$.

Three or more points in a plane are said to be collinear if all of them lie on the same straight line.

(i) Points A, B and C marked on a single straight line are collinear.

(ii) Points A, B and C which do not lie on the same straight line are non-collinear. Taking the points two at a time, we get the pairs (A, B), (B, C) and (C, A). Through each pair exactly one line can be drawn, giving the lines AB, BC and CA.

Hence, three different lines can be drawn through three non-collinear points, namely line AB, line BC and line CA.

(i) Collinear points lie on the same straight line, so a single line already passes through all three of them.

Hence, only one line can be drawn passing through three collinear points.

(ii) Taking the three collinear points A, B and C two at a time, we get the pairs (A, B), (B, C) and (A, C). Each pair determines one line segment.

Hence, three line segments are determined, namely $\mathbf{\overline{AB}, \overline{BC} \text{ and } \overline{AC}}$.

Three or more lines in a plane are said to be concurrent if all of them pass through the same point. This common point is called the point of concurrence of the given lines.

In the figure below, the four lines all pass through the point P, so ${\overleftrightarrow{AB}}, {\overleftrightarrow{CD}}, {\overleftrightarrow{EF}}, {\overleftrightarrow{GH}}$are concurrent and P is their point of concurrence.

(i) The figure has four collinear points P, Q, R and S. Taking them two at a time, the line segments are $\overline{PQ}, \overline{QR}, \overline{RS}, \overline{PR}, \overline{PS} \text{ and } \overline{QS}$.

∴ Number of line segments = 6.

(ii) The line segments in the figure are:

$\overline{AB}, \overline{AD}, \overline{AG}, \overline{AE}, \overline{AC}, \overline{GE}, \overline{GC}, \overline{EC}, \overline{GH}, \overline{BH}, \overline{BE}, \overline{BD}, \overline{BC}, \overline{HE}, \overline{HD}, \overline{ED} \text{ and } \overline{CD}$.

∴ Number of line segments = 17.

(iii) The line segments in the figure are:

$\overline{EF}, \overline{EH}, \overline{EP}, \overline{FG}, \overline{FP}, \overline{FH}, \overline{GP} \text{ and } \overline{GH}$.

∴ Number of line segments = 8.

(i) A line segment has two end points.

(ii) A line has no end points.

(iii) A ray has one end point.

(iv) A line segment has a definite length.

(v) A line AB is represented by $\mathbf{\overleftrightarrow{AB}}$.

(vi) A ray AB is represented by $\mathbf{\overrightarrow{AB}}$.

(vii) A point shows a definite position.

(viii) Two lines intersect in a point.

(ix) Two planes intersect in a line.

(x) The minimum number of points of intersection of three lines in a plane is 0.

(xi) The maximum number of points of intersection of three lines in a plane is 3.

(i) Parallel lines are lines which do not intersect even when produced.

∴ The pairs of parallel lines are (l, m) and (p, q).

(ii) Intersecting lines are lines which cross each other, and the common point is their point of intersection.

∴ The pairs of intersecting lines and their points of intersection are:

l and p at C; l and q at A; l and n at A; m and n at B; m and p at B; m and q at D; n and p at B; n and q at A.

(iii) Collinear points are three or more points lying on the same straight line.

∴ The sets of collinear points are (A, B, E), (A, C, H), (D, B, K), (C, B, F) and (A, D, G).

(iv) Concurrent lines are three or more lines passing through the same point.

∴ Lines l, n and q are concurrent at A; lines m, n and p are concurrent at B.

(v) Three non-collinear points are three points which do not lie on the same straight line.

∴ One set of three non-collinear points is A, C and D.

(i) The wheel of a bicycle has the shape of a circle.

(ii) A blackboard has the shape of a rectangle.

(iii) A page of a book has the shape of a rectangle.

(iv) The face of the full moon has the shape of a circle.

(v) A set-square in a geometry box has the shape of a triangle.

(vi) A face of a dice has the shape of a square.

(i) A triangle has 3 sides and 3 vertices.

(ii) A rectangle has 4 sides and 4 vertices.

(iii) All the sides of a square are equal.

(iv) The opposite sides of a rectangle are equal.

(v) A circle has no side and no vertex.

From the figure, the points lie on the line in the order A, C, D, E, B.

(i) False — Ray EB starts at E and extends in the direction of B. The point D lies on the opposite side of E, so D is not a point on ray EB.

(ii) True — Ray DA starts at D and extends in the direction of A. Since C lies between D and A, the point C is on ray DA.

(iii) True — Ray DB has initial point D while ray EB has initial point E. As their initial points differ, the two rays are different.

(iv) True — The line segment CE has end points C and E, and D lies between them, so C, D and E all lie on segment CE.

(v) True — Ray CA starts at C towards A while ray CB starts at C towards B (opposite direction). They are opposite rays, hence different.

(i) False — A ray extends endlessly in one direction, so it does not have a finite length.

(ii) False — Ray AB has initial point A and ray BA has initial point B, and they extend in opposite directions, so they are different rays.

(iii) True — A line has no end points and can be named from either direction, so line AB is the same as line BA.

(iv) True — Two distinct points fix the two end points, and hence determine exactly one line segment.

(v) False — An unlimited (infinite) number of lines can be drawn through a single given point.

(vi) False — The points A, B and C lie on one straight line and the points C, D and E lie on one straight line, but the two sets have only the single point C in common. Two straight lines can meet in at most one point, so they need not be the same line.

For example, let A, B and C lie on a vertical line and let C, D and E lie on the horizontal line drawn through C. Each set is collinear, yet A, B, D and E do not all lie on one straight line.

A ray has exactly one end point, called its initial point, and extends endlessly in one direction.

Hence, option 1 is the correct option.

A line segment is the straight path between two fixed points, so it has two end points.

Hence, option 2 is the correct option.

A line extends endlessly in both directions, so it has no end points.

Hence, option 3 is the correct option.

A ray and a line extend endlessly, so their lengths cannot be measured. A line segment has two fixed end points and therefore a definite, measurable length.

Hence, option 3 is the correct option.

A ray and a line extend endlessly, and a plane extends indefinitely in all directions, so none of these can be drawn completely on paper. Only a line segment, having a definite length, can be drawn fully on a piece of paper.

Hence, option 2 is the correct option.

Through a single given point, lines can be drawn in every possible direction, so an infinite number of lines can pass through it.

Hence, option 4 is the correct option.

Two distinct points fix a unique direction, so exactly one line can be drawn passing through them.

Hence, option 1 is the correct option.

Two lines in a plane, if not parallel, cross each other at one common point.

Hence, option 1 is the correct option.

The minimum number of points of intersection occurs when all three lines are parallel to one another. Parallel lines never meet, so there are 0 points of intersection.

Hence, option 4 is the correct option.

The maximum number of points of intersection occurs when no two of the three lines are parallel and no three lines pass through the same point. In that case each pair of lines meets at a distinct point, giving 3 points of intersection.

Hence, option 3 is the correct option.

From a given point, a ray can be drawn in every possible direction, so infinitely many rays can be drawn with that point as their initial point.

Hence, option 3 is the correct option.

Two distinct points are the two end points of the straight path joining them, so they determine exactly one line segment.

Hence, option 1 is the correct option.

(i) A line segment has a definite length.

(ii) A line segment has two end points.

(iii) A ray has one end point.

(iv) A line has no end point.

(v) Two lines may intersect only at one point.

(i) False — Only one line can be drawn passing through two given points.

(ii) False — A ray extends endlessly in one direction, so it has no definite length.

(iii) True — An infinite number of lines can be drawn through a single given point.

(iv) True — Three lines in a plane meet at the most at three points of intersection.

(v) True — A line extends endlessly in both directions, so it cannot be drawn completely on paper.

Assertion (A) is true, because through any two points exactly one straight line can be drawn, so two points always lie on a common line and are therefore collinear.

Reason (R) is false, because points are collinear only when they lie on the same straight line, not merely on the same plane.

Hence, option 3 is the correct option.

Assertion (A) is true, because a line segment is the straight path between two fixed points, so it has two end points.

Reason (R) is true, because a line segment has a definite length and two end points, while a line has no end points and no definite length.

But Reason (R) only states that a line segment and a line are different, it does not explain why a line segment has two end points.

Hence, option 2 is the correct option.

Assertion (A) is false, because parallel lines never meet even when produced in either direction.

Reason (R) is true, because the distance between two parallel lines always remains the same throughout.

Hence, option 4 is the correct option.

Since $\overline{AB} = \overline{BC} = \overline{CD} = \overline{DE} = 6$ cm.

The points lie in order A, B, C, D, E, so:

$\overline{BE} = \overline{BC} + \overline{CD} + \overline{DE} = 6 + 6 + 6 = 18$ cm.

Calculating,

$$\begin{aligned}\Rightarrow 2\overline{BC} + \overline{BE} - 3\overline{CD}\\[1em] \Rightarrow (2 \times 6) + 18 - (3 \times 6)\\[1em] \Rightarrow 12 + 18 - 18\\[1em] \Rightarrow 12 \text{ cm }.\end{aligned}$$

Hence, option 1 is the correct option.

Two lines are parallel if they never meet, however far they are produced. So we look for lines running in the same direction and count them two at a time.

In the given figure there are two horizontal lines AB and CD, and four vertical lines EF, GH, IJ and KL.

The two horizontal lines AB and CD are parallel to each other. Taking them together, we get:

(AB, CD) → 1 pair

The four vertical lines EF, GH, IJ and KL are all parallel to one another. Taking them two at a time, we get:

(EF, GH), (EF, IJ), (EF, KL), (GH, IJ), (GH, KL), (IJ, KL) → 6 pairs

A horizontal line and a vertical line cut each other, so they are not parallel. Such pairs are not counted.

∴ Total pairs of parallel lines = 1 + 6 = 7.

Hence, option 3 is the correct option.

The four line segments shown in the figure are EF, IJ, KL and GH.

The pairs of intersecting line segments are:

EF and KL intersect at A. EF and IJ intersect at D. KL and IJ intersect at B. KL and GH intersect at C. GH and IJ intersect at E.

Thus, there are 5 pairs of intersecting line segments.

Hence, option 3 is the correct option.