krit.club logo

Constructions and Tilings - Geometric Constructions

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Construction of a line parallel to a given line, through a point not on the line, using the concept of equal alternate interior angles or corresponding angles.

•

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side: a+b>ca + b > c.

•

SSS Criterion: Construction of a triangle when the lengths of its three sides are known.

•

SAS Criterion: Construction of a triangle when the lengths of two sides and the measure of the angle between them (included angle) are known.

•

ASA Criterion: Construction of a triangle when the measures of two angles and the length of the side included between them are known.

•

RHS Criterion: Construction of a right-angled triangle when the length of one leg and its hypotenuse are known.

•

Tilings (Tessellations): A pattern of shapes that fits together perfectly without any gaps or overlaps to cover a plane.

📐Formulae

Triangle Inequality Property: a+b>c\text{Triangle Inequality Property: } a + b > c

Angle Sum Property: ∠A+∠B+∠C=180∘\text{Angle Sum Property: } \angle A + \angle B + \angle C = 180^\circ

Pythagoras Property (for RHS): a2+b2=c2\text{Pythagoras Property (for RHS): } a^2 + b^2 = c^2

💡Examples

Problem 1:

Construct a triangle ΔABC\Delta ABC given AB=5 cmAB = 5\text{ cm}, BC=6 cmBC = 6\text{ cm}, and AC=7 cmAC = 7\text{ cm}.

Solution:

  1. Draw a line segment BC=6 cmBC = 6\text{ cm}. 2. With BB as centre and radius 5 cm5\text{ cm}, draw an arc. 3. With CC as centre and radius 7 cm7\text{ cm}, draw another arc intersecting the previous arc at AA. 4. Join ABAB and ACAC.

Explanation:

This follows the SSS (Side-Side-Side) construction criterion where all three sides are specified.

Problem 2:

Can a triangle be constructed with sides 3 cm3\text{ cm}, 4 cm4\text{ cm}, and 8 cm8\text{ cm}?

Solution:

Check the triangle inequality: 3+4=73 + 4 = 7. Since 7<87 < 8, the sum of two sides is not greater than the third side.

Explanation:

According to the triangle inequality property a+b>ca + b > c, a triangle cannot be formed because 3+4≯83 + 4 \ngtr 8.

Problem 3:

Construct a right-angled triangle ΔLMN\Delta LMN where ∠M=90∘\angle M = 90^\circ, MN=3 cmMN = 3\text{ cm}, and LN=5 cmLN = 5\text{ cm}.

Solution:

  1. Draw MN=3 cmMN = 3\text{ cm}. 2. At MM, draw a ray MXMX perpendicular to MNMN (using a protractor or compass). 3. With NN as centre and radius 5 cm5\text{ cm}, draw an arc cutting MXMX at LL. 4. Join LNLN.

Explanation:

This uses the RHS (Right-angle Hypotenuse Side) criterion. LNLN is the hypotenuse (5 cm5\text{ cm}) and MNMN is the given side (3 cm3\text{ cm}).