Solutions: Section 2.4 (Points, Lines, Rays)

Figure it Out (Page 15)

Q1. Can you help Rihan and Sheetal find their answers?

• Rihan: Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?
• Sheetal: Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points?

Solution:

• Rihan’s Answer: Rihan can draw many (uncountable) number of lines through a single given point.
• Sheetal’s Answer: Sheetal can draw only one unique line through the two given points.

Exercise Questions (Page 16)

Q2. Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?

Solution:

• Line Segments: $\overline{LM}, \overline{MP}, \overline{PQ}, \overline{QR}$.
• Points on exactly one segment: Points $L$ and $R$.
• Points on two segments: Points $M, P$, and $Q$.

Q3. Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?

Solution:

• Rays: $\overrightarrow{TA}, \overrightarrow{TB}, \overrightarrow{TN}, \overrightarrow{NB}$.
• Is T the starting point?
  • No. $T$ is the starting point for $\overrightarrow{TB}, \overrightarrow{TN}$, and $\overrightarrow{TA}$.
  • $T$ is not the starting point for $\overrightarrow{NB}$ (Start point is $N$).

Q4. Draw a rough figure and write labels appropriately to illustrate each of the following: a. $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ meet at $O$. b. $\overleftrightarrow{XY}$ and $\overleftrightarrow{PQ}$ intersect at point $M$. c. Line $l$ contains points $E$ and $F$ but not point $D$. d. Point $P$ lies on $\overline{AB}$.

Solution: (Refer to visual descriptions below based on standard geometric conventions)

• (a) Two rays starting from a common point $O$.
• (b) Two lines crossing each other like an ‘X’ at point $M$.
• (c) A straight line with dots $E$ and $F$ on it, and a dot $D$ floating outside the line.
• (d) A line segment with ends $A$ and $B$, and a dot $P$ somewhere on the line between them.

Q5. In Fig. 2.6, name: a. Five points b. A line c. Four rays d. Five line segments

Solution:

• a. Five points: $D, E, O, B, C$.
• b. A line: $\overleftrightarrow{DE}$ or $\overleftrightarrow{DO}$ or $\overleftrightarrow{DB}$ or $\overleftrightarrow{EO}$ or $\overleftrightarrow{EB}$ or $\overleftrightarrow{OB}$. (They represent the same line).
• c. Four rays: $\overrightarrow{OC}, \overrightarrow{OB}, \overrightarrow{OE}, \overrightarrow{OD}$.
• d. Five line segments: $\overline{DE}, \overline{DO}, \overline{DB}, \overline{EO}, \overline{EB}$. (Others like $\overline{OB}, \overline{OC}$ are also possible).

Q6. Here is a ray OA (Fig. 2.7). It starts at O and passes through the point A. It also passes through the point B. a. Can you also name it as $\overrightarrow{OB}$? Why? b. Can we write $\overrightarrow{OA}$ as $\overrightarrow{AO}$? Why or why not?

Solution:

• a. Yes. Since $O$ is the starting point and point $B$ lies on the ray that goes endlessly in the direction of $A$, $\overrightarrow{OB}$ represents the same ray as $\overrightarrow{OA}$. $\overrightarrow{OA}$ is an extension of $\overrightarrow{OB}$.
• b. No. $\overrightarrow{OA}$ is a ray starting at $O$ and going towards $A$. $\overrightarrow{AO}$ would represent a ray starting at $A$ and going towards $O$. They have different starting points and directions.