Practice Questions

Name the two fundamental tools that are used for basic geometric constructions.

Define the term 'perpendicular bisector'.

To construct an angle of $22.5^\circ$ using only a ruler and compass, which two geometric constructions must be applied in sequence? Analyze the steps.

What is meant by 'tiling' in geometry?

Analyze whether a standard $7 \times 7$ grid can be tiled perfectly by $2 \times 1$ dominoes. Justify your answer by calculating the area.

Demonstrate the construction of a regular hexagon with a side length of 4 cm. After constructing the hexagon, construct its circumcircle (the circle passing through all its vertices).

Describe the meaning of 'bisection' in geometry and provide one example for a line segment and one for an angle.

Explain the key property of points lying on a perpendicular bisector which is used for its construction.

Justify why the standard compass and ruler construction for a $60^\circ$ angle is geometrically correct.

Identify the angle that is constructed when you create an equilateral triangle.

Explain the relationship between a regular hexagon and equilateral triangles that helps in the construction of the hexagon.

What is the main principle involving angles that allows for the construction of a line parallel to another given line?

Demonstrate the construction of a triangle ABC where base AB = 7 cm, $\angle CAB = 45^\circ$, and $\angle CBA = 60^\circ$. Use only a ruler and compass and show all construction arcs.

Explain the step-by-step procedure to construct the perpendicular bisector of a line segment XY. Also, state the geometric reason why this method is accurate.

To construct the perpendicular bisector of a line segment PQ of length 10 cm, what is the minimum radius you must set on your compass? Analyze and explain why.

When constructing a line parallel to a given line through an external point, the method involves copying an angle. Analyze which pair of angles (e.g., corresponding, alternate interior) are constructed to be equal.

Summarize the information given in the text about the Śulba-Sūtras. What were these texts, what was their main purpose, and what primary tool was used for the constructions described in them?

Demonstrate the steps to construct an angle of $135^\circ$ using only a ruler and compass. Label the final angle.

Two circles with centers P and Q intersect at points A and B. The radius of both circles is 6 cm. Analyze the geometric relationship between the line segment PQ (joining the centers) and the common chord AB. Justify your analysis.

Demonstrate how to find a point P on a given line segment AB of length 8 cm such that AP = $\frac{1}{4}$ AB, using only a ruler and compass. List the steps.

Draw a line m and a point K not on it. First, construct a line n through K parallel to m. Second, construct a line p that is perpendicular to m and passes through K. Analyze the relationship between lines n and p.

Demonstrate the construction of a design consisting of a square of side 6 cm with four semi-circles drawn on each side as a diameter, where the semi-circles point outwards from the square. Show all construction lines.

A student claims they can construct a perpendicular bisector of a line segment XY by drawing arcs of radius 5 cm from X and arcs of radius 6 cm from Y, both above and below the line. Critique this method. Is it valid? Justify your answer.

Formulate a condition that must be met for an $m \times n$ rectangular grid to be tileable by $1 \times 3$ tiles (trominoes).

Propose a method to construct a $135^\circ$ angle using only a ruler and compass. You do not need to perform the construction, only outline the steps.

Design a geometric pattern for a circular region. Describe the steps to create your design using only a ruler and compass, starting with a given circle.

The text describes a method to construct a line parallel to a given line by copying an angle. Justify why this construction is guaranteed to produce a parallel line by referencing the properties of parallel lines and transversals.

Create a set of instructions to construct a rhombus with diagonals of length 8 cm and 6 cm using only a ruler and compass. Justify why your construction results in a rhombus.

List two examples of regular polygons that can be used to tile the entire plane without any gaps.

Formulate and justify a method to construct a $75^\circ$ angle using only a ruler and compass.

Create a step-by-step guide to construct a regular hexagon inscribed in a circle of a given radius 'r'. Justify why the side length of the resulting hexagon is precisely equal to the radius of the circle.

Design and justify a method to divide a given line segment AB into four equal parts using only a ruler and compass.

Describe the first two steps required to construct a $90^\circ$ angle at a given point 'O' on a straight line.

Summarize the method for bisecting a given angle, $\angle PQR$, using a compass and a ruler.

Design a complex geometric pattern that is constructible using only a ruler and compass. The design must start with a single line segment and must incorporate at least three of the following constructions: perpendicular bisector, angle bisection, construction of a $60^\circ$ angle, and construction of a parallel line. Provide a clear, step-by-step description of how to create your design.

Describe in detail the steps to copy a given angle, $\angle PQR$, onto a new ray, MN. Explain the geometric principle that guarantees the new angle is congruent to the original.

An L-shaped tromino is a tile made of three $1 \times 1$ squares. Analyze if a full $8 \times 8$ grid can be tiled by L-shaped trominoes. Then, analyze if an $8 \times 8$ grid with any single $1 \times 1$ square removed can be tiled by L-shaped trominoes. Justify your reasoning.

A tiler has an unlimited supply of identical, regular pentagonal tiles. Can they tile an entire flat floor without any gaps or overlaps? Evaluate this possibility and justify your reasoning.

Explain the chessboard coloring argument that can be used to show that a grid with an odd number of squares cannot be tiled by $2 \times 1$ dominoes.

A regular hexagon can be divided into six congruent triangles by joining its opposite vertices. Analyze and identify the specific type of these triangles.

A $6 \times 6$ chessboard has two opposite corner squares removed (e.g., top-left and bottom-right). Analyze if the remaining 34 squares can be tiled by $2 \times 1$ dominoes using the coloring method.

Given an acute angle $\angle PQR$. Demonstrate the steps to construct a new angle $\angle XYZ$ such that $\angle XYZ = 2 \times \angle PQR$ using only a ruler and compass.

A standard $8 \times 8$ chessboard has two squares removed from opposite corners. Is it possible to tile the remaining 62 squares with $2 \times 1$ dominoes? Evaluate the problem and provide a justification for your conclusion.

A student attempts to bisect $\angle ABC$. First, they draw an arc from B, cutting AB at P and BC at Q. Then, they set their compass to the length PQ and draw intersecting arcs from P and Q to find a point R. They claim the line BR bisects the angle. Critique this method. Is it always correct?

The ancient Śulba-Sūtras used a rope to construct a perpendicular bisector. A rope of fixed length is looped onto pegs at points X and Y. Its midpoint is then pulled taut to a point A above XY, and then to a point B below XY. Formulate a proof using triangle congruence to justify that the line AB is the perpendicular bisector of XY.