Introduction

Geometric construction is a foundational topic in Class 10 Mathematics under the West Bengal Board (WBBSE). In Chapter 21, students learn about determining the mean proportional between two given line segments using only a compass and straightedge. This chapter demonstrates a core geometrical technique without relying on numerical calculations or algebraic formulas. Instead, the entire solution depends on accurate, logical steps drawn with precision. The process is deeply rooted in classical Euclidean geometry and reinforces key ideas such as similarity and right triangles.

What is Mean Proportional?

In geometry, the mean proportional between two positive real numbers, say aaa and bbb, is a value x such that:

ax= xb or x2=ab

Thus, the mean proportional is the square root of the product of the two values:

x=ab

 

When this concept is applied geometrically, the task becomes constructing a line segment whose length is exactly the mean proportional between two given lengths.

Geometrical Interpretation

In this chapter, instead of solving the equation algebraically, students are guided to construct the mean proportional between two given line segments, using geometrical tools:

• A straightedge (scale without markings) 
• A compass 

The geometric construction is based on the property of right-angled triangles, specifically the Altitude Theorem from Euclidean geometry.

Key Concept:

In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse creates two smaller triangles that are similar to each other and to the original triangle. This leads to the proportionality:

Altitude2=(Segment 1)×(Segment 2)

 

This is the geometric basis for finding the mean proportional.

Step-by-Step Construction of Mean Proportional

Objective:

To construct a line segment equal in length to the mean proportional between two given segments AB and BC, i.e., find BD such that:

BD2 = AB × BC

Let’s assume the lengths of AB and BC are known.

Step 1: Draw a straight line

• Use a scale to draw a line and mark points A and C such that the total length AC=AB+BC
• Place point B between A and C such that AB and BC are the two given segments.
  

Step 2: Construct a semicircle with AC as diameter

• Using a compass, find the midpoint O of the segment AC.
  
• With center O and radius OA (or OC), draw a semicircle passing through points A and C.
  

Step 3: Erect a perpendicular from point B

• From point B, construct a perpendicular line to AC that intersects the semicircle at point D.
  

Step 4: Join B and D

• Connect point B to point D. The line segment BD is the required mean proportional between AB and BC.
  

Justification of the Construction

The triangle ADC is a right-angled triangle, since any triangle inscribed in a semicircle with the diameter as its base is a right triangle (by the Thales’ Theorem).

Here, triangle ABD and triangle CBD are similar by AA (Angle-Angle) criterion. Therefore, the following proportion holds:

ABBD = BDBC ⇒ BD2=AB×BC⇒BD= AB × BC

Hence, BD is the mean proportional geometrically constructed.

Example (Using Specific Lengths)

Let’s assume:

• AB=4 cm
• BC=9 cm

Then,

• AC=13 cm

Follow the same steps:

1. Draw line AC=13 cm
2. Mark B such that AB=4 cm
3. Draw semicircle on AC
4. Erect perpendicular from B
5. Intersect at D and join BD

Then,

BD=AB × BC = 4 × 9 = 36 =6cm

 

So, the constructed segment BD is exactly 6 cm – the mean proportional.

Why Is This Chapter Important?

• Reinforces understanding of similar triangles, right angles, and semicircle properties. 
• Helps develop visual reasoning and geometrical accuracy. 
• Forms a bridge between algebraic concepts (mean proportional) and geometric construction. 
• Strengthens foundation for higher classes involving circle geometry and coordinate geometry.
  

Conclusion

Chapter 21 – Construction: Determination of Mean Proportional – is a vital geometry topic in the Class 10 WBBSE syllabus. It combines logic, construction skill, and geometric theory to solve a classical problem using only a compass and straightedge. The construction process not only illustrates mathematical beauty but also enhances precision and confidence in practical geometry.