HOTS Questions: Introduction to Euclid’s Geometry

Q1: Show that : length AH > sum of lengths of AB + BC + CD.

Sol:
We have
AH = AB + BC + CD + DE + EF + FG + GH
Each of the segments DE, EF, FG and GH has a positive length, so AB + BC + CD is only a part of the whole sum AH.
Therefore the whole AH is greater than the part AB + BC + CD.
⇒ AH > AB + BC + CD
Hence, length AH > sum of lengths AB + BC + CD.

Q2: For given four distinct points in a plane, find the number of lines that can be drawn through :
(i) When all four points are collinear.
(ii) When three of the four points are collinear.
(iii) When no three of the four points are collinear.

Sol:

(i) Consider the points given are A, B, C and D.
When all the four points are collinear :
One line only, because all points lie on the same straight line. Any two of these points determine the same single line.

(ii) When three of the four points are collinear :
Four lines in all.
Explanation: Take the three collinear points as A, B and C and the fourth point as D. The line through A, B and C is one line. Joining D to each of A, B and C gives three more distinct lines. Hence total distinct lines = 1 + 3 = 4.
Here, we have four lines

 (Four)

(iii) When no three of the four points are collinear :

Six lines.

Explanation: If no three points are collinear, every pair of points determines a distinct line. Number of pairs of four points = C(4,2) = 6, so there are 6 distinct lines.

Here, we have

 (Six)

Q3: Three lighthouse towers are located at points A, B and C on the section of a national forest to protect animals from hunters by the forest department as shown in figure. Which value is department exhibiting by locating extra towers ? How many straight lines can be drawn from A to C? State the Euclid Axiom which states the required result. Give one more. Postulate.

Sol: By locating extra towers the department is showing concern for wildlife conservation and public responsibility. This action reflects care for animals and protection of the environment.
How many straight lines can be drawn from A to C?
One and only one straight line can be drawn from A to C. Any two distinct points determine a single straight line between them.
Euclid axiom that states the required result:

Euclid's First Postulate: "A straight line may be drawn from any point to any other point."
One more postulate:

Euclid's Third Postulate: "A circle may be described with any centre and any radius."
These postulates are basic rules used to construct and reason about straight lines and figures in plane geometry.

Q4: Rohan's maid has two children of same age. Both of them have equal number of dresses. Rohan on his birthday plans to give both of them same number of dresses. What can you say about the number of dresses each one of them will have after Rohan's birthday? Which Euclid's axiom is used to answer this question? What value is Rohan depicting by doing so ? Write one more Euclid's axiom.
Sol: Here, the two children start with an equal number of dresses, and Rohan gives the same number of dresses to each child.
Therefore, after the birthday both children will still have equal numbers of dresses.
Reason using Euclid's axiom:

Euclid's Axiom 2: If equals are added to equals, then the wholes are equal. This justifies that adding the same number to equal quantities keeps them equal.
Value depicted by Rohan: fairness and kindness (he treats both children equally and shows care).

One more Euclid's axiom:
Euclid's Axiom 3: If equals are subtracted from equals, then the remainders are equal.