CBSE Class 8 Mathematics Practical Geometry Worksheet Set 04

Question 1. Construct a quadrilateral ABCD in which AB = 4.4cm, BC = 4cm, CD = 6.4cm, DA = 2.8cm and BD = 6.6cm
Answer: 1. Draw a line segment BD = 6.6 cm.
2. With B as center and radius 4.4 cm, draw an arc.
3. With D as center and radius 2.8 cm, draw an arc intersecting the previous arc at point A.
4. Join AB and AD.
5. With B as center and radius 4 cm, draw an arc on the other side of BD.
6. With D as center and radius 6.4 cm, draw an arc intersecting this arc at point C.
7. Join BC and DC.
\(\text{ABCD}\) is the required quadrilateral.
In simple words: A quadrilateral with four sides and one diagonal can be constructed by splitting it into two triangles.

Exam Tip: Always draw a rough sketch first to understand the placement of sides and diagonals correctly.

Question 2. Construct a parallelogram ABCD where AB = 3.6cm, BC = 4.2cm and AC = 6.5cm.
Answer: 1. Draw a line segment AB = 3.6 cm.
2. With A as center and radius 6.5 cm, draw an arc.
3. With B as center and radius 4.2 cm, draw an arc intersecting the previous arc at point C.
4. Join AC and BC.
5. With A as center and radius 4.2 cm, draw an arc.
6. With C as center and radius 3.6 cm, draw an arc intersecting the previous arc at point D.
7. Join AD and CD.
\(\text{ABCD}\) is the required parallelogram.
In simple words: Opposite sides of a parallelogram are equal, so \(\text{CD} = \text{AB}\) and \(\text{AD} = \text{BC}\).

Exam Tip: Use the property that opposite sides of a parallelogram are equal in length to determine missing dimensions.

Question 3. Construct a rhombus with side 6cm and one diagonal 8cm. Measure the other diagonal.
Answer: 1. Draw a diagonal AC = 8 cm.
2. With A as center and radius 6 cm, draw an arc above AC and another arc below AC.
3. With C as center and radius 6 cm, draw arcs intersecting the previous arcs at points B and D.
4. Join AB, BC, CD, and DA.
5. Measure the diagonal BD using a ruler. The length of the other diagonal BD is approximately \(8.9\) cm.
In simple words: All four sides of a rhombus are equal, allowing us to construct it like two triangles sharing a diagonal.

Exam Tip: Remember that all sides of a rhombus are equal in length.

Question 4. Construct a quadrilateral ABCD in which AB = 5.5cm, AD = 4.4cm, CD = 6.5cm, AC = 6.5cm and BD = 7.1cm.
Answer: 1. Draw a line segment AB = 5.5 cm.
2. With A as center and radius 6.5 cm (AC), draw an arc.
3. With B as center and radius 7.1 cm (BD), we cannot directly intersect without another point, so we use triangle ABD first: draw arc from A with radius 4.4 cm (AD) and from B with radius 7.1 cm (BD) to get point D.
4. Join AD and BD.
5. From A, draw an arc of radius 6.5 cm (AC). From D, draw an arc of radius 6.5 cm (CD) to intersect at point C.
6. Join AC, BC, and DC.
\(\text{ABCD}\) is the required quadrilateral.
In simple words: Construct triangle ABD first, then locate point C using distances from A and D.

Exam Tip: Identify which three measurements form a solvable triangle first to start your construction.

Question 5. Construct a rhombus COLD in which CL = 7.5cm and OD = 6cm.
Answer: 1. Draw diagonal CL = 7.5 cm.
2. Draw the perpendicular bisector of CL intersecting it at point O' (center).
3. Since diagonals of a rhombus bisect each other at right angles, cut arcs of length \(\frac{6}{2} = 3\) cm above and below the bisector along the perpendicular line.
4. Mark the intersection points as D and OD = 6 cm total diagonal, so distance from center is 3 cm.
5. Join CO, OL, LD, and DC.
\(\text{COLD}\) is the required rhombus.
In simple words: Diagonals of a rhombus bisect each other perpendicularly.

Exam Tip: When two diagonals are given, draw one diagonal first, construct its perpendicular bisector, and mark half the length of the second diagonal on both sides.

Question 6. Construct a rectangle PURE in which PU = 5.5cm, UR = 4cm
Answer: 1. Draw line segment PU = 5.5 cm.
2. At points P and U, construct \(90^\circ\) angles using a compass or set square.
3. With U as center, draw an arc of radius 4 cm along the perpendicular line to locate point R.
4. With P as center, draw an arc of radius 4 cm along the perpendicular line to locate point E.
5. Join RE.
\(\text{PURE}\) is the required rectangle.
In simple words: Opposite sides of a rectangle are equal, and each interior angle is \(90^\circ\).

Exam Tip: Always construct accurate \(90^\circ\) angles at the vertices of a rectangle.

Question 7. Construct a parallelogram HARD in which HA = 7cm, AR = 5cm, \(\angle D = 105^\circ\).
Answer: 1. Draw line segment HA = 7 cm.
2. Adjacent angles of a parallelogram are supplementary, so \(\angle A = 180^\circ - 105^\circ = 75^\circ\).
3. At point A, construct an angle of \(75^\circ\).
4. With A as center and radius AR = 5 cm, draw an arc on the \(75^\circ\) ray to locate point R.
5. Since opposite sides are equal, with R as center and radius 7 cm, draw an arc, and with H as center and radius 5 cm, draw an arc to intersect at point D.
6. Join RD and HD.
\(\text{HARD}\) is the required parallelogram.
In simple words: Consecutive angles in a parallelogram add up to \(180^\circ\).

Exam Tip: Use the supplementary property of adjacent angles in a parallelogram to find missing angle measures.

Question 8. Construct a quadrilateral BIRD where BI = 3.5cm, IR = 6.5cm, \(\angle B = 75^\circ\), \(\angle I = 105^\circ\) and \(\angle R = 120^\circ\).
Answer: 1. Draw line segment BI = 3.5 cm.
2. At point I, construct an angle of \(105^\circ\).
3. With I as center and radius IR = 6.5 cm, draw an arc along the ray to locate point R.
4. At point R, construct an angle of \(120^\circ\).
5. At point B, construct an angle of \(75^\circ\).
6. Extend the rays from B and R until they intersect at point D.
\(\text{BIRD}\) is the required quadrilateral.
In simple words: When two adjacent sides and three angles are known, construct base angles and use side lengths to find vertices.

Exam Tip: Ensure angle rays are drawn long enough before finding the intersection point for the final vertex.

Question 9. Construct a quadrilateral PQRS, in which \(\angle Q = 45^\circ\), \(\angle R = 90^\circ\), QR = 5cm, PQ = 9cm and RS = 7cm.
Answer: 1. Draw line segment QR = 5 cm.
2. At point Q, construct an angle of \(45^\circ\).
3. With Q as center and radius PQ = 9 cm, draw an arc on the \(45^\circ\) ray to locate point P.
4. At point R, construct an angle of \(90^\circ\).
5. With R as center and radius RS = 7 cm, draw an arc on the \(90^\circ\) ray to locate point S.
6. Join PS.
\(\text{PQRS}\) is the required quadrilateral.
In simple words: Draw the included side first, make the given angles at its endpoints, and mark off the given side lengths.

Exam Tip: Double-check angle construction steps for \(45^\circ\) (bisector of \(90^\circ\)) and \(90^\circ\).

Question 10. Construct a kite SOLD if OD = 8cm, SD = 5cm and LD = 6cm. Which properties of the kite did you use in the process.
Answer: 1. Draw diagonal OD = 8 cm.
2. With D as center and radius 5 cm, draw arcs on both sides of OD.
3. With O as center and radius 6 cm, draw arcs intersecting the previous arcs at points S and L.
4. Join OS, SD, OL, and LD.
\(\text{SOLD}\) is the required kite.
Properties used: In a kite, two pairs of consecutive sides are equal in length (\(\text{OS} = \text{OL}\) and \(\text{SD} = \text{LD}\)).
In simple words: A kite has two pairs of equal adjacent sides, allowing construction using compass arcs.

Exam Tip: Clearly state the geometric property of equal adjacent sides when asked.

Question 11. How will you construct a rectangle PLOT if you know only the lengths PL and LO?
Answer: 1. Draw line segment PL equal to the given length.
2. Construct a \(90^\circ\) angle at point L.
3. With L as center and radius equal to LO, draw an arc along the perpendicular ray to locate point O.
4. Since opposite sides of a rectangle are equal and parallel, \(\text{PT} = \text{LO}\) and \(\text{OT} = \text{PL}\).
5. With O as center and radius equal to PL, draw an arc. With P as center and radius equal to LO, draw an arc to intersect at point T.
6. Join OT and PT to complete rectangle PLOT.
In simple words: Two adjacent sides of a rectangle are sufficient because opposite sides are equal and all angles are \(90^\circ\).

Exam Tip: Mention that adjacent lengths in a rectangle implicitly give all four side lengths due to opposite side equality.

Question 12. Construct a square GOAT with GO = 4.9cm
Answer: 1. Draw line segment GO = 4.9 cm.
2. At points G and O, construct \(90^\circ\) angles.
3. With G as center and radius 4.9 cm, draw an arc along the perpendicular at G to locate point T.
4. With O as center and radius 4.9 cm, draw an arc along the perpendicular at O to locate point A.
5. Join TA.
\(\text{GOAT}\) is the required square.
In simple words: All four sides of a square are equal and all interior angles are \(90^\circ\).

Answer: 1. Draw diagonal \(d_1 = 6.6\) cm.
2. Draw the perpendicular bisector of this diagonal, intersecting it at the midpoint.
3. Take half of the second diagonal: \(\frac{4.8}{2} = 2.4\) cm.
4. With the midpoint as center, cut arcs of radius 2.4 cm on both sides along the perpendicular bisector.
5. Join the endpoints of both diagonals to form the rhombus.
In simple words: A rhombus can be uniquely constructed when both of its diagonals are given by using their perpendicular intersection property.

Exam Tip: Always divide the second diagonal length by 2 before marking distances from the center.

Question 14. Construct a square ABCD, given that diagonal AC = 6cm.
Answer: 1. Draw diagonal AC = 6 cm.
2. Draw the perpendicular bisector of AC, intersecting AC at point O.
3. Since diagonals of a square are equal and bisect each other at right angles, the distance from O to B and O to D is also equal to half of AC, which is \(\frac{6}{2} = 3\) cm.
4. Cut arcs of radius 3 cm on the perpendicular bisector above and below point O to locate B and D.
5. Join AB, BC, CD, and DA.
\(\text{ABCD}\) is the required square.
In simple words: Diagonals of a square are equal and perpendicular bisectors of each other.

Exam Tip: Use the property that square diagonals bisect each other at \(90^\circ\) and are equal in length.

Question 15. Construct a parallelogram CARE where CA = 3.9cm and AR = 5.5cm.
Answer: Note: To construct a unique parallelogram, either an angle or a diagonal is also required along with two adjacent sides. Assuming standard supplementary data or completing with a given angle if implied, but with only two adjacent sides given, infinitely many parallelograms can be formed. Assuming standard problem format where two adjacent sides and included angle (or diagonal) are provided, we construct using adjacent sides CA = 3.9 cm, AR = 5.5 cm, and assuming an angle or diagonal if specified, or construct with general angle \(\theta\):
1. Draw CA = 3.9 cm.
2. Construct a convenient angle at A (e.g., \(60^\circ\) or as per standard school problem assumption).
3. Cut AR = 5.5 cm along the ray.
4. Complete the parallelogram using opposite sides \(\text{CE} = \text{AR}\) and \(\text{RE} = \text{CA}\).
In simple words: Parallelograms require at least three independent measurements (like two adjacent sides and an angle).

Exam Tip: Verify that sufficient independent parameters are provided before constructing quadrilaterals.

Question 16. Construct a rectangle with adjacent lengths 3.5cm and 4.5cm
Answer: 1. Draw a line segment of length 4.5 cm.
2. Construct \(90^\circ\) angles at both endpoints of the line segment.
3. Cut arcs of length 3.5 cm along both perpendicular rays.
4. Join the top endpoints to complete the rectangle.
In simple words: A rectangle is fully determined by its two adjacent side lengths since all angles are \(90^\circ\).

Exam Tip: Use a compass or protractor to ensure precise right angles at the corners.

Question 17. Construct a rhombus ABCD, with side AB = 7.3cm and \(\angle A = 75^\circ\).
Answer: 1. Draw line segment AB = 7.3 cm.
2. At point A, construct an angle of \(75^\circ\).
3. With A as center and radius 7.3 cm, draw an arc on the \(75^\circ\) ray to locate point D.
4. Since all sides of a rhombus are equal, with D as center and radius 7.3 cm, draw an arc, and with B as center and radius 7.3 cm, draw an arc to intersect at point C.
5. Join DC and BC.
\(\text{ABCD}\) is the required rhombus.
In simple words: A rhombus has all sides equal, so knowing one side and one angle is enough for construction.