Mock Test : Based on Ganita Prakash | Class 8 | Chapter 4 (QUADRILATERALS)

Figure it Out - Mock Test

1. Awesome—let’s build a “Figure it Out” mock test modeled on the Grade 8 style from page 94. We’ll keep the focus on geometry reasoning, construction, and properties, with clear, varied question types. You can print this and use it as a mini test or practice sheet.

   • ∠PQO

   • ∠POQ

   • ∠QOR

   • ∠ORS

   • And all other angles formed by the sides and diagonals.

2. In the rhombus PQRS shown below, the diagonals intersect at point O. If ∠OPQ=40∘, find all the other angles inside the rhombus.

3. You are told that a quadrilateral has diagonals that are perpendicular to each other.

   • (i) What two special types of quadrilaterals could this be?
   • (ii) If you are then told that the diagonals also bisect each other, what must the figure be?
   • (iii) If you are instead told that only one diagonal bisects the other, what must the figure be?

4. The midpoints of the sides of a square are joined in order to form a new quadrilateral. What special type of quadrilateral is this new figure? Give a reason for your answer. (Hint: Think about the small triangles you cut off at the corners).

5. You have built a four-sided gate. You measure the opposite sides and find they are equal in length, which tells you it is a parallelogram. However, you want to ensure it is a perfect rectangle and not leaning to one side. What single, simple measurement can you take with a tape measure to confirm it is a true rectangle? Explain why this measurement works.

6. We know that one of the properties of a square is that its diagonals are equal and perpendicular. Can this be chosen as the definition of a square? In other words, is every quadrilateral with equal and perpendicular diagonals a square? (Hint: Try drawing one.)

Figure it Out — Mock Test (Grade 8)

1. Angles in Rectangles
   a) In rectangle ABCD, diagonals AC and BD intersect at O. If ∠OAB = 25°, find all interior angles formed at O and at the vertices inside the triangles AOB, BOC, COD, and DOA.
   b) In rectangle PQRS, diagonals intersect at M. If ∠QMR = 100°, find ∠PMQ, ∠RMS, and the base angles of triangles QMR and PMS.

2. Constructing Quadrilaterals with Given Diagonal Conditions
   Draw quadrilaterals whose diagonals:
   a) Are equal (8 cm each), bisect each other, and intersect at 60°.
   b) Are equal (8 cm each), bisect each other, and intersect at 90°.
   c) Are equal (8 cm each), bisect each other, and intersect at 120°.
   For each, name the type of quadrilateral obtained and justify your answer briefly.

3. Circle and Diameters Puzzle
   A circle has center O. PR and QS are two diameters that intersect at O. Let points A be the midpoint of arc PQ (not containing R), and B be the midpoint of arc RS (not containing Q).
   a) What type of quadrilateral is AQOB? Explain your reasoning.
   b) Are the diagonals of AQOB equal, perpendicular, or both? Justify.

4. Make an Exact Right Angle (No Paper)
   You have two equal sticks and a piece of thread.
   a) Describe a method to create an exact 90° angle using only these items.
   b) Explain why your method guarantees a right angle (reference a geometric property).

5. Definitions and Properties
   a) Can “opposite sides parallel and equal” be used as the complete definition of a rectangle? Explain and provide a counterexample if necessary.
   b) State two different properties that, when added to “parallelogram,” are sufficient to define a rectangle.

6. Bonus: Rhombus vs Rectangle
   a) A quadrilateral has equal diagonals that bisect each other. Is it necessarily a rectangle? Explain.
   b) A quadrilateral has all sides equal and diagonals that intersect at right angles. What is it? Are its diagonals necessarily equal? Explain.

7. Challenge—Angle Chase in a Parallelogram
   In parallelogram KLMN, diagonals intersect at O. If ∠KON = 70° and ∠LOM = 110°, find ∠OKL and ∠OMN. Show reasoning using isosceles triangle properties formed by the diagonals.

8. Construction Task—Perpendicular Bisector via Thread
   Using only a thread and a pencil:
   a) Describe how to construct the perpendicular bisector of a given segment AB.
   b) Explain why this construction works (use the idea of equal radii circles or loci).

9. Identify the Quadrilateral
   For each set of diagonal properties, identify the most specific quadrilateral:
   a) Diagonals are equal and bisect each other.
   b) Diagonals bisect each other and are perpendicular.
   c) Diagonals are equal, bisect each other, and are perpendicular.
   d) Diagonals bisect each other but are not equal and not perpendicular.
   Briefly justify each.

10. Reasoning with Midpoints
    In rectangle XYZW, let M and N be the midpoints of sides XY and ZW, respectively.
    a) Prove or disprove: MN is parallel to XZ and YW.
    b) Find the length of MN in terms of the rectangle’s length and width, and interpret the result geometrically.

Answer Key Outline (for teacher/parent use)

• Q1: Use rectangle properties: diagonals equal and bisect; triangles around intersection are isosceles; sum of angles in triangle; vertical and linear pairs.
• Q2: 60°/120° give rectangles (not squares). 90° gives a square. All: equal diagonals + bisect → parallelogram; equal diagonals → rectangle; plus perpendicular → square (when sides equal by symmetry).
• Q3: AQOB is a kite or rectangle depending on arc definitions; typically, using diameter mid-arc points yields right angles at O; discuss symmetry of circle and equal chords.
• Q4: Methods include using the thread to form a rectangle by equal-stick diagonals or creating a Thales semicircle: forming a right angle in a semicircle.
• Q5: “Opposite sides parallel and equal” defines a parallelogram, not necessarily a rectangle. Sufficient conditions: parallelogram with one right angle; or parallelogram with equal diagonals.
• Q6: a) Equal, bisecting diagonals → rectangle (for parallelograms), but not every quadrilateral; need additional conditions. b) Rhombus; diagonals perpendicular but not equal unless square.
• Q7: Use isosceles triangles from equal halves of diagonals; angle relations at O determine base angles.
• Q8: Classic perpendicular bisector by two circles with equal radius using thread; intersection points create equal distances from A and B; line through intersections is the perpendicular bisector.
• Q9: a) Rectangle (in a parallelogram context). b) Rhombus or kite (if not assuming parallelogram); rhombus if also bisect. c) Square (under parallelogram context). d) General parallelogram.
• Q10: MN is parallel to XZ and YW; MN equals the rectangle’s width (or length) depending on orientation; can be shown via midpoint theorem or coordinate geometry.