Construction : Determination of Mean Proportional

Multiple Choice Questions (MCQ)

1.The theory of natural selection was primarily proposed by:
(a) Jean-Baptiste Lamarck
(b) Gregor Mendel
(c) Charles Darwin
(d) Hugo de Vries
Answer: (c) Charles Darwin

2.Structures in different organisms that have a similar underlying anatomical structure but different functions are called:
(a) Analogous organs
(b) Homologous organs
(c) Vestigial organs
(d) Fossils
Answer: (b) Homologous organs

3. Which of the following is an example of adaptation for camouflage?
(a) Brightly colored feathers of a peacock
(b) Long neck of a giraffe
(c) Green color of a grasshopper
(d) Sharp claws of a lion
Answer: (c) Green color of a grasshopper

4. The sudden heritable changes in the genetic material of an organism are known as:
(a) Variations
(b) Adaptations
(c) Mutations
(d) Speciation
Answer: (c) Mutations

5. The process by which new species arise from existing ones is called:
(a) Evolution
(b) Adaptation
(c) Variation
(d) Speciation
Answer: (d) Speciation

Short Answer Questions (SAQ)

1. What is meant by the mean proportional between two line segments AB and BC?
Answer: It is a line segment BD such that BD² = AB × BC, or BD = √(AB × BC).

2. Which construction step ensures that triangle ADC is a right triangle?
Answer: Drawing a semicircle on AC as diameter ensures triangle ADC is right-angled at D (by Thales’ Theorem).

3. What are the two main instruments used in this construction?
Answer: A compass and a straightedge (ruler without markings).

Long Answer Questions (LAQ)

1. Explain the steps to geometrically construct the mean proportional between two given segments AB and BC.
Answer: Step 1: Draw a straight line AC = AB + BC and mark point B such that AB and BC are the two segments.
Step 2: Find midpoint O of AC and draw a semicircle with diameter AC.
Step 3: From point B, erect a perpendicular that intersects the semicircle at point D.
Step 4: Join B and D.
Conclusion: BD is the mean proportional between AB and BC.

2. Justify geometrically why the constructed segment BD is the mean proportional.
Answer:

Triangle ADC is right-angled (Thales’ Theorem).

Triangles ABD and CBD are similar (AA criterion).

Using the similarity property:
AB / BD = BD / BC ⇒ BD² = AB × BC

Therefore, BD = √(AB × BC), which is the required mean proportional.

Construction : Determination of Mean Proportional