Constructions - Free MCQ Practice Test with solutions, Class 8 Mathematics

To construct a rhombus given one side and one angle, you begin by drawing the specified side. From one endpoint, you construct the given angle to mark the next side. Then, using the same side length, you draw arcs from both endpoints to find the other two corners of the rhombus. This method highlights the properties of rhombuses, where all sides are equal in length.

To bisect a given angle, you start by drawing an arc from the vertex that intersects both arms of the angle. Next, you take the intersection points and draw arcs of equal radius from each of them, which will intersect at a point inside the angle. Joining this point with the angle's vertex gives you the bisector, effectively dividing the angle into two equal parts. This method is foundational in geometric constructions and illustrates the concept of angle division.

When constructing a parallelogram with two consecutive sides and one diagonal, the first step is to draw the diagonal. From one endpoint of this diagonal, you then draw arcs for the lengths of the consecutive sides to find the other vertices. This method emphasizes the geometric properties of parallelograms, particularly how the diagonals relate to the sides.

When constructing a rectangle given two adjacent sides, you first draw one side of the rectangle. At one endpoint, you construct a 90° angle to establish the second side. After marking the length of the second side, you can then use arcs to find the opposite corners and complete the rectangle. This method emphasizes the properties of rectangles, particularly the right angles at each vertex.

To construct a square when given two adjacent sides, you first draw one side to the specified length. From one endpoint, you construct a 90° angle to establish the second side. After marking the length of the second side, you can then use arcs to find the remaining corners of the square. This construction demonstrates the importance of right angles and equal side lengths in defining squares.

To create a perpendicular line from a point outside a given line, you begin by drawing arcs from the external point that intersect the line at two points. From these intersection points, you then draw arcs of equal radius that meet on the opposite side of the line. Joining the external point to this intersection creates the desired perpendicular line. This construction demonstrates the principles of perpendicularity and how to effectively use a compass and straightedge in geometric constructions.

To construct a parallelogram when two consecutive sides and the included angle are known, you start by drawing one side. From the endpoint of this side, you use the angle to draw the next side. After marking the length of the second side, you can then use arcs to find the necessary intersection points to complete the parallelogram. This construction relies on the fact that opposite sides of a parallelogram are equal in length and parallel.

To construct a rhombus when both diagonals are given, you first draw one diagonal and find its midpoint by constructing the perpendicular bisector. From this midpoint, you then measure half the length of the other diagonal on both sides to determine the other vertices of the rhombus. Joining these points completes the rhombus. This method highlights how the properties of diagonals and symmetry are essential in constructing rhombuses.

To construct a 45° angle, you first create a 90° angle using standard methods, such as drawing a line and using arcs. Once you have the 90° angle, you then bisect this angle to obtain the 45° angle. This technique is important because it illustrates how angles can be subdivided, showcasing the principles of angle bisection in geometry.

To draw a perpendicular from a point on a line, you start by using the given point as the center to draw an arc that intersects the line at two points. From these points, you then draw arcs of equal radius which intersect at a point above or below the line. Joining the original point to this intersection point creates the perpendicular line. This construction emphasizes the geometric principles of perpendicularity without the need for physical angle measurement.

To construct a diagonal in a rectangle when one side is known, you first draw the side. At one endpoint, you create a 90° angle to establish the second side. Once you have the dimensions of the rectangle, the diagonal can be easily drawn by connecting the opposite corners. This method reinforces understanding the relationships in rectangles, including how the diagonals bisect each other.

To construct a square when one side is given, start by drawing the given side. At one endpoint, you construct a 90° angle, marking the second side. Finally, by using the same length as the first side, you can find the other vertices of the square using arcs. This process illustrates how squares are defined by equal sides and right angles.

To construct the perpendicular bisector of a line segment, you draw arcs from both endpoints of the segment using a radius greater than half the length of the segment. The arcs will intersect at two points. Joining these intersection points creates a line that bisects the segment at a right angle, effectively creating the perpendicular bisector. This method illustrates the concept of symmetry and the properties of bisectors in geometry.

To construct a 60° angle, you begin by drawing a line segment. Then, with one endpoint as the center, you draw an arc that intersects the segment. Using this intersection as the center and keeping the same radius, you draw another arc to intersect the first arc. Joining the first endpoint to this intersection point forms the desired 60° angle. This construction relies on the properties of equilateral triangles, where each angle measures 60°.

To construct a line parallel to an existing line at a specified distance, you first draw a perpendicular line from a point on the original line. From this point, you then measure the desired distance along the perpendicular and mark a new point. Finally, you draw another line through this new point that is perpendicular to the first perpendicular line, creating a parallel line at the specified distance. This method illustrates the properties of parallel lines and how they can be constructed using perpendiculars.

To construct a line parallel to a given line through a specific point, you first join the point to any point on the given line. Then, you construct an angle at the given point that is equal to the angle formed at the point on the line. Extending the arm of this angle creates a parallel line. This method highlights the concept of alternate angles and their properties in parallel line constructions.

When constructing a rectangle with one side and one diagonal known, you start by drawing the given side. At one endpoint, you create a right angle, and then from the other endpoint, you draw an arc using the length of the diagonal to intersect the line at the right angle. This intersection helps determine the other vertices of the rectangle. This method emphasizes the relationships between sides and diagonals in rectangular constructions.

To construct an angle equal to a given angle, you start by drawing a line segment as the base for the new angle. Then, using the vertex of the given angle as the center, you draw an arc that intersects both arms of the angle. By measuring the distance between the intersection points with a compass and replicating this distance from the vertex of the new angle, you can accurately create an angle equal to the original. This method emphasizes the use of geometric properties rather than physical measurements.

To construct a 30° angle, you first create a 60° angle, as the 30° angle is half of 60°. After constructing the 60° angle using the described method, you then bisect it to achieve the desired 30° angle. This technique demonstrates the relationship between angles and the concept of angle bisectors, which is critical in geometric constructions.

To construct a quadrilateral with four sides and one angle, you begin by drawing one side. At one endpoint, you construct the specified angle, marking the second side. Then, from the other endpoint of the second side, you draw arcs with the lengths of the remaining sides to find the intersection points, which complete the quadrilateral. This method emphasizes the relationship between sides and angles in quadrilateral constructions.