GSEB Class 6 Maths Solutions Chapter 14 Practical Geometry InText Questions

GSEB Solutions

Try These (Page 286)

Question 1. In Step II of the construction using ruler and compasses, (NCERT Textbook page 285) what would happen if we take the length of radius to be smaller than half the length of \( \overline{AB} \)?
Answer: If we choose a radius that is smaller than half of the length of \( \overline{AB} \), the arcs will not cross each other at the two points P and Q. This prevents the successful construction of the desired geometric shape.
In simple words: If the drawing tool's opening is too small, the curved lines won't meet, stopping the drawing process.

Exam Tip: Remember that the radius chosen for drawing arcs in constructions must always be greater than half the length of the line segment to ensure the arcs intersect.

Try These (Page 289)

Question 1. In Step II (See NCERT Page 289) above, what would happen if we take radius to be smaller than half the length BC?
Answer: If we select a radius that is smaller than half of the length of BC, the arcs drawn with centres B and C will not intersect each other. This means the construction will not be possible or accurate.
In simple words: Making the compass radius too short means the arcs from points B and C won't cross paths.

Exam Tip: For constructing perpendicular bisectors or angles, selecting an appropriate radius (greater than half the segment length) is crucial for arc intersection.

Try These (Page 290)

Question 1. How will you construct a 15° angle?
Answer: The construction of a \( 15^\circ \) angle involves several clear steps:
Steps of construction:
Step I: First, construct an angle \( ABC \) measuring \( 60^\circ \). This is the foundation.
Step II: Next, bisect \( \angle ABC \) to obtain an angle of \( 30^\circ \). This new angle is labeled \( \angle ABD = 30^\circ \).
Step III: Finally, bisect \( \angle ABD \). Let \( \overrightarrow{BE} \) be the bisector of \( \angle ABD \). Thus, \( \angle ABE = \frac{1}{2}(30^\circ) = 15^\circ \). This completes the angle construction.
In simple words: To make a \( 15^\circ \) angle, first draw a \( 60^\circ \) angle. Then, cut that \( 60^\circ \) angle in half to get \( 30^\circ \). After that, cut the \( 30^\circ \) angle in half again, and you'll have \( 15^\circ \).

Exam Tip: Constructing specific angles often requires bisecting larger angles (like \( 60^\circ \) to get \( 30^\circ \), then \( 30^\circ \) to get \( 15^\circ \)). Practice these basic bisections thoroughly.

Question 1. How will you construct a 150° angle?
Answer: Here are the steps to construct a \( 150^\circ \) angle:
Steps of construction:
Step I: Start by drawing a straight line labeled 'l' and mark a point O on it.
Step II: Using O as the center and a convenient radius, draw an arc that intersects 'l' at point A.
Step III: Keep the same radius and, with A as the center, draw an arc to cut the first arc at point B.
Step IV: Again, maintaining the same radius and using B as the center, draw another arc to intersect the first arc at point C.
Step V: Once more, with the same radius and using C as the center, draw an arc to cut the first arc at point D.
Step VI: Now, bisect \( \angle COD \) in such a way that \( \angle COE = \angle EOD = 30^\circ \). This division is crucial.
Since \( 150^\circ = 120^\circ + 30^\circ \), by combining these angle parts, we find that \( \angle AOC + \angle COE = \angle AOE \). Thus, \( \angle AOE \) is the required angle, whose measure is \( 150^\circ \).
In simple words: To draw a \( 150^\circ \) angle, start with a line and point O. Mark off \( 60^\circ \) sections to reach \( 120^\circ \). Then, bisect the next \( 60^\circ \) section to add \( 30^\circ \), making a total of \( 150^\circ \).

Exam Tip: Angles like \( 150^\circ \) are often constructed by combining standard angles (like \( 60^\circ \) and \( 90^\circ \)) or their bisections. Knowing how to construct \( 60^\circ \), \( 90^\circ \), and \( 120^\circ \) is fundamental.

Try These (Page 291)

Question 1. How will you construct a 45° angle?
Answer: To construct a \( 45^\circ \) angle, follow these construction steps:
Steps of construction:
Step I: First, construct an angle of \( 90^\circ \). As the figure demonstrates, \( \angle POQ = 90^\circ \). This right angle is the starting point.
Step II: Next, draw the angle bisector OR for \( \angle POQ \). This bisector splits the \( 90^\circ \) angle into two equal parts. So, \( \frac{1}{2}(\angle POQ) = \frac{1}{2}(90^\circ) = 45^\circ \). Therefore, \( \angle POQ = 45^\circ \), which is the desired angle.
In simple words: To make a \( 45^\circ \) angle, first draw a perfect right angle, which is \( 90^\circ \). Then, simply cut that \( 90^\circ \) angle exactly in half. Each half will be \( 45^\circ \).

Exam Tip: The construction of a \( 45^\circ \) angle directly follows from bisecting a \( 90^\circ \) angle. Ensure your \( 90^\circ \) angle construction is accurate for a precise \( 45^\circ \) result.