Constructions & Tessellations MCQ Test — Class 7 CBSE

Solution:

The method of constructing a parallel line often involves drawing a transversal and then copying an angle at a different position.

If the alternate interior angles formed by a transversal with two lines are equal, then the lines are parallel. This is the geometric principle applied in this construction method.

Solution:

After drawing the base segment PQ, the next step is to locate the third vertex R.

Vertex R is 3 cm away from Q (QR = 3 cm), so an arc with center Q and radius 3 cm is a valid first arc to draw.

Similarly, vertex R is 7 cm away from P (RP = 7 cm), so an arc with center P and radius 7 cm would also be a valid first arc. The intersection of these two arcs will give point R.

Solution:

Apply the triangle inequality theorem to each set of side lengths. For a triangle to be formed, the sum of any two sides must be greater than the third side.

A) 2 + 2 = 4, which is not > 5. Cannot form a triangle.

B) 4 + 5 = 9 > 8; 4 + 8 = 12 > 5; 5 + 8 = 13 > 4. All conditions are met. Can form a triangle.

C) 3 + 4 = 7, which is not > 7. Cannot form a triangle.

D) 1 + 2 = 3, which is not > 3. Cannot form a triangle.

Solution:

The given information is Side-Angle-Side (SAS): AB, ∠B, and BC.

First, draw the base segment AB = 6 cm.

The included angle is ∠B, which is at point B. Therefore, the next step is to construct an angle of 60° at point B, by drawing a ray BX such that ∠ABX = 60°.

Solution:

The acronym SAS stands for Side-Angle-Side.

For this criterion to construct a unique triangle, the angle specified must be precisely the one that lies between the two given sides. This is known as the included angle.

Solution:

Let's review the criteria for constructing a unique triangle: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and RHS (Right-angle-Hypotenuse-Side for right triangles).

A) Two sides and a non-included angle (SSA) is an ambiguous case and does not always lead to a unique triangle.

B) Three angles (AAA) only determines the shape of the triangle, not its size, so many similar triangles can be formed (not unique).

C) The perimeter alone does not provide enough information to construct a unique triangle.

D) The lengths of all three sides (SSS criterion) ensures the construction of a unique triangle, provided the triangle inequality theorem is satisfied.

Solution:

Geometric constructions, especially for drawing parallel lines by copying angles (either corresponding or alternate interior angles), primarily rely on specific tools.

A ruler is needed to draw straight lines and connect points.

A compass is essential for drawing arcs to mark equal distances and to accurately copy angles without using degree measurements.

Solution:

A tessellation, also commonly known as tiling, is a pattern of shapes that fit together perfectly to cover a surface.

The key characteristics of a tessellation are that the shapes must cover an entire surface (they often repeat) and there should be no gaps or overlaps between them.

Solution:

For a regular polygon to tessellate a plane on its own, its interior angle must be an exact divisor of 360° (because multiple copies of the polygon must meet at a point without gaps or overlaps).

Interior angle of an equilateral triangle = (3-2)×180°/3 = 60°. 360° is divisible by 60° (360/60 = 6).

Interior angle of a regular pentagon = (5-2)×180°/5 = 108°. 360° is not divisible by 108°.

Interior angle of a regular octagon = (8-2)×180°/8 = 135°. 360° is not divisible by 135°.

Interior angle of a regular heptagon = (7-2)×180°/7 ≈ 128.57°. Not a divisor of 360°.

Therefore, only the equilateral triangle (along with squares and regular hexagons) can form a regular tessellation on its own.

Solution:

For shapes to tile a surface without any gaps or overlaps, the vertices of the shapes must meet perfectly at a common point.

When multiple angles meet at a single vertex and completely surround it, their sum must form a full circle.

The sum of angles around a point in a plane is always 360°.