GSEB Class 8 Maths Solutions Chapter 4 પ્રાયોગિક ભૂમિતિ Exercise 4.1

Gujarat Board Textbook Solutions Class 8 Maths Chapter 4 પ્રાયોગિક ભૂમિતિ Ex 4.1

Question 1. Construct the following quadrilaterals:
(i) Quadrilateral ABCD
\( \overline{\mathrm{AB}} = 4.5 \) cm
\( \overline{\mathrm{BC}} = 5.5 \) cm
\( \overline{\mathrm{CD}} = 4 \) cm
\( \overline{\mathrm{AD}} = 6 \) cm
\( \overline{\mathrm{AC}} = 7 \) cm
Answer:

Construction Steps:
1. Draw a line segment \( \overline{\mathrm{AB}} \) that measures \( 4.5 \) cm.
2. With point A as the center, and a compass radius of \( 7 \) cm, draw an arc.
3. Then, with point B as the center, and a compass radius of \( 5.5 \) cm, draw another arc that cuts the first arc. Mark the point where these two arcs meet as C.
4. Next, with point A as the center, and a compass radius of \( 6 \) cm, draw a third arc.
5. After that, with point C as the center, and a compass radius of \( 4 \) cm, draw a fourth arc that cuts the third arc. Mark the point where these two arcs meet as D.
6. Connect the points by drawing line segments \( \overline{\mathrm{BC}} \), \( \overline{\mathrm{CD}} \), \( \overline{\mathrm{AD}} \), and \( \overline{\mathrm{AC}} \).
7. The shape ABCD is the desired quadrilateral.
In simple words: First, draw a line for side AB. Then, use a compass from points A and B to find point C. Next, use a compass from points A and C to find point D. Finally, join all the points with lines to complete the quadrilateral.

Exam Tip: When constructing a quadrilateral with given sides and a diagonal, always begin by drawing the diagonal first to form two triangles, which simplifies the construction process.

(ii) Quadrilateral JUMP
\( \overline{\mathrm{JU}} = 3.5 \) cm
\( \overline{\mathrm{UM}} = 4 \) cm
\( \overline{\mathrm{MP}} = 5 \) cm
\( \overline{\mathrm{PJ}} = 4.5 \) cm
\( \overline{\mathrm{PU}} = 6.5 \) cm
Answer:

Construction Steps:
1. Draw a line segment \( \overline{\mathrm{JU}} \) that measures \( 3.5 \) cm.
2. With point J as the center, and a compass radius of \( 4.5 \) cm, draw an arc.
3. Then, with point U as the center, and a compass radius of \( 6.5 \) cm, draw another arc that cuts the first arc. Mark the point where these two arcs meet as P.
4. Next, with point U as the center, and a compass radius of \( 4 \) cm, draw a third arc.
5. After that, with point P as the center, and a compass radius of \( 5 \) cm, draw a fourth arc that cuts the third arc. Mark the point where these two arcs meet as M.
6. Connect the points by drawing line segments \( \overline{\mathrm{JP}} \), \( \overline{\mathrm{UM}} \), \( \overline{\mathrm{MP}} \), and \( \overline{\mathrm{UP}} \).
7. The shape JUMP is the desired quadrilateral.
In simple words: Start by drawing JU. Use a compass from J and U to find P using given lengths. Then, from U and P, find M using their lengths. Lastly, connect all the vertices.

Exam Tip: For quadrilaterals with two triangles, constructing one triangle first (e.g., JUP) using three side lengths makes it easier to complete the second triangle (UMP).

(iii) Parallelogram MORE
\( \overline{\mathrm{RE}} = 4.5 \) cm
\( \overline{\mathrm{EO}} = 7.5 \) cm
(Note: In a parallelogram, opposite sides are equal, so \( \overline{\mathrm{MO}} = \overline{\mathrm{RE}} = 4.5 \) cm and \( \overline{\mathrm{ME}} = \overline{\mathrm{OR}} = 6 \) cm, as shown in the figure.)
Answer:

Explanation:
MORE is a parallelogram. In a parallelogram, the lengths of its opposite sides are equal.
Therefore, \( \overline{\mathrm{RE}} = \overline{\mathrm{MO}} = 4.5 \) cm, and \( \overline{\mathrm{OR}} = \overline{\mathrm{ME}} = 6 \) cm.
Construction Steps:
1. Draw a line segment \( \overline{\mathrm{MO}} \) that measures \( 4.5 \) cm.
2. With point M as the center, and a compass radius of \( 6 \) cm, draw an arc.
3. Then, with point O as the center, and a compass radius of \( 7.5 \) cm, draw another arc that cuts the first arc. Mark the point where these two arcs meet as E.
4. Next, with point O as the center, and a compass radius of \( 6 \) cm, draw a third arc.
5. After that, with point E as the center, and a compass radius of \( 4.5 \) cm, draw a fourth arc that cuts the third arc. Mark the point where these two arcs meet as R.
6. Connect the points by drawing line segments \( \overline{\mathrm{ME}} \), \( \overline{\mathrm{OR}} \), \( \overline{\mathrm{RE}} \), and \( \overline{\mathrm{OE}} \).
7. The shape MORE is the desired parallelogram.
In simple words: First, draw the base MO. Using the diagonal EO and side ME, locate point E. Then, using side OR and RE, locate point R. Finally, connect all the points to form the parallelogram. Remember opposite sides are equal.

Exam Tip: When constructing a parallelogram, remember that opposite sides are equal in length. Use this property to find missing side lengths before starting your construction.

(iv) Rhombus BEST
\( \overline{\mathrm{BE}} = 4.5 \) cm
\( \overline{\mathrm{ET}} = 6 \) cm
Answer:

Explanation:
BEST is a rhombus. All four sides of a rhombus have the same length.
Therefore, \( \overline{\mathrm{BE}} = \overline{\mathrm{ES}} = \overline{\mathrm{ST}} = \overline{\mathrm{TB}} = 4.5 \) cm, and the diagonal \( \overline{\mathrm{ET}} = 6 \) cm.
Construction Steps:
1. Draw a line segment \( \overline{\mathrm{BE}} \) that measures \( 4.5 \) cm.
2. With point B as the center, and a compass radius of \( 4.5 \) cm, draw an arc.
3. Then, with point E as the center, and a compass radius of \( 6 \) cm, draw another arc that cuts the first arc. Mark the point where these two arcs meet as T.
4. Next, with point E as the center, and a compass radius of \( 4.5 \) cm, draw a third arc.
5. After that, with point T as the center, and a compass radius of \( 4.5 \) cm, draw a fourth arc that cuts the third arc. Mark the point where these two arcs meet as S.
6. Connect the points by drawing line segments \( \overline{\mathrm{BT}} \), \( \overline{\mathrm{ES}} \), \( \overline{\mathrm{ST}} \), and \( \overline{\mathrm{ET}} \).
7. The shape BEST is the desired rhombus.
In simple words: Draw side BE first. Use the diagonal ET and side BT from B to find T. Then, from E and T, find S. Finally, connect all the vertices to complete the rhombus, remembering all sides are the same length.

Exam Tip: For rhombus construction, always utilize the property that all four sides are of equal length. This reduces the number of unique measurements you need to track.