Unit 11: Construction (रचना) Notes & Steps
Introduction
Unit 11: Construction (रचना) applies theoretical theorems of area to construct geometric figures of equal area. The primary focus is transforming triangles into parallelograms and quadrilaterals into triangles or parallelograms with specific dimensions.
1. Theoretical Basis (सैद्धान्तिक आधारहरू)
Construction in this unit relies on three fundamental area theorems:
-
Theorem 1: Parallelograms standing on the same base and between the same parallel lines are equal in area.
($Area\ of\ ||^{gm} = Area\ of\ ||^{gm}$) -
Theorem 2: The area of a triangle is half the area of a parallelogram standing on the same base and between the same parallel lines.
($Area\ of\ \Delta = \frac{1}{2} \times Area\ of\ ||^{gm}$) -
Theorem 3: Triangles standing on the same base and between the same parallel lines are equal in area.
($Area\ of\ \Delta = Area\ of\ \Delta$)
2. Construction of Parallelogram equal to a Triangle
Question Pattern:
Construct a parallelogram having area equal to a given triangle ABC and having one angle $x^\circ$.
Steps of Construction:
- Construct the given Triangle ABC with specified measurements.
- Bisect the base BC at point D (so that $BD = DC$).
- Through vertex A, draw a line $PQ || BC$.
- At point D, construct the given angle $\angle CDE = x^\circ$ such that E lies on line PQ.
- Cut off $EF = DC$ on the parallel line PQ.
- Join C and F. Then, CDEF is the required parallelogram.
Reason: Area of $\Delta ABC$ = Area of $||^{gm} CDEF$ (Since both are on the same base DC (half of BC) and between same parallels).
3. Construction of Triangle equal to a Quadrilateral
[Image of Quadrilateral to Triangle Construction Diagram]Question Pattern:
Construct a triangle equal in area to a given quadrilateral ABCD.
Steps of Construction:
- Construct the Quadrilateral ABCD with given dimensions.
- Join diagonal DB.
- From vertex C, draw a line $CE || DB$ to meet the extended line AB at point E.
- Join D and E.
- Now, $\Delta ADE$ is the required triangle whose area is equal to Quadrilateral ABCD.
4. Construction of Parallelogram equal to a Quadrilateral
Question Pattern:
Construct a parallelogram equal in area to a quadrilateral ABCD.
Steps of Construction:
- Step 1 (Quad $\to$ Triangle): First, construct the quadrilateral ABCD and convert it into an equal area $\Delta ADE$ (as shown in Type 2).
- Step 2 (Triangle $\to$ Parallelogram): Now, convert the $\Delta ADE$ into a parallelogram (as shown in Type 1).
- Bisect the base AE of the new triangle at point M.
- Draw required angle at M and complete the parallelogram on the base AM (or ME).
Full Chapter PDF Manual
For detailed geometric drawings and examples, access the complete PDF manual below.
Disclaimer: This content is based on the Class 10 Compulsory Mathematics manual by Dr. Simkhada (Readmore Publishers). Please refer to the PDF for exact measurements and construction arcs.