CBSE Class 9 Mathematics Introduction to Euclid's Geometry MCQs Set 13

Answer: (a) Infinitely many

Explanation:
1. Any direction through the point gives a different line.
2. So infinitely many lines pass through a given point.

Teacher's Note:
1. A single point does not fix a direction.
2. Two points are needed to fix one line.

Answer: (b) one point

Explanation:
1. If two distinct lines had two common points, both would be the unique line through those points.
2. So two distinct lines have at most one common point.

Teacher's Note:
1. This is Theorem 5.1 of the chapter.
2. Parallel lines have no common point.

Answer: (c) one line

Explanation:
1. The common points of two intersecting planes form a straight line.
2. Think of two pages of a book meeting at the spine.

Teacher's Note:
1. Planes can also be parallel and have no common point.
2. The meeting set is a line, not a point.

Answer: (d) \( 3 non-collinear points \)

Explanation:
1. Three collinear points lie on infinitely many planes.
2. Three points not on one line lie on exactly one plane.

Teacher's Note:
1. Three non-collinear points fix a plane just as two points fix a line.
2. The condition non-collinear is essential.

Answer: (a) \( 3 \)

Explanation:
1. Each segment needs two end points: \( AB,\ BC,\ AC \).
2. So there are 3 segments.

Teacher's Note:
1. Count the pairs of points: 3 choose 2.
2. Four collinear points would give 6.

Answer: (b) \( 16 \)

Explanation:
1. \( AC=AB+BC \), so \( AB=10-6=4 \).
2. \( AB^2=16 \).

Teacher's Note:
1. B lies on the segment, so the lengths add.
2. Square the length at the end.

Answer: (c) Euclid

Explanation:
1. Euclid collected and organised the geometry of his time in his book Elements, using axioms and postulates.

Teacher's Note:
1. Euclid's Elements was the standard geometry text for centuries.
2. The method of proof from axioms starts with him.

Answer: (d) False, because infinitely many lines pass through any point

Explanation:
1. Through any point we can draw lines in every direction.
2. So the statement is false.

Teacher's Note:
1. Compare with 'a point lies on infinitely many lines'.
2. Exactly one line passes through two distinct points, not through one.

Answer: (a) True, by Euclid's second postulate

Explanation:
1. A terminated line is a line segment.
2. Postulate 2 says a segment can be extended on either side to form a line.

Teacher's Note:
1. Extending a segment gives the whole line.
2. The postulate allows extension both ways.

Answer: (b) False, there are infinitely many such circles

Explanation:
1. A circle needs a centre and a radius.
2. Different radii give different circles with the same centre.

Teacher's Note:
1. Postulate 3 allows any centre and any radius.
2. Concentric circles share a centre.

Answer: (c) False, because dividing by 2 gives x = y

Explanation:
1. \( 2x=2y \) gives \( x=y \) by dividing equals by 2.
2. So distinct x and y cannot satisfy it.

Teacher's Note:
1. Euclid's axiom: halves of equal things are equal.
2. Test with numbers: 2 times 1 is not 2 times 2.

Answer: (d) they are collinear

Explanation:
1. By Playfair's axiom only one line through \( A \) is parallel to \( l \).
2. So \( AB,\ AC,\ AD,\ AE \) are the same line and the points lie on it.

Teacher's Note:
1. Through a point outside a line there is exactly one parallel to it.
2. All the named lines coincide.

Answer: (a) \( 1 \)

Explanation:
1. If all four are collinear, the unique line containing them is the only one.
2. Any two of the points determine the same line.

Teacher's Note:
1. Two distinct points fix one line.
2. Collinear means on a single line.

Answer: (b) \( 4 \)

Explanation:
1. Let \( A,B,C \) be collinear and \( D \) off the line.
2. The line \( ABC \) is one line; \( D \) joined to \( A \), to \( B \) and to \( C \) gives three more.
3. In total there are 4 lines.

Teacher's Note:
1. Count the common line once, then count the lines through the outside point.
2. Not 3, since D can be joined to each of A, B and C by a different line.

Answer: (c) \( 6 \)

Explanation:
1. Each pair of points gives a line and no two pairs give the same line.
2. \( \binom42=6 \) lines.

Teacher's Note:
1. Collinearity would reduce the count.
2. Count pairs: AB, AC, AD, BC, BD, CD.

Answer: (d) Infinitely many

Explanation:
1. All the points lie on one line.
2. Infinitely many planes contain a given line, like pages of a book on the spine.

Teacher's Note:
1. Three non-collinear points fix a unique plane.
2. A line alone does not fix a plane.

Answer: (a) Exactly one

Explanation:
1. Three points not on a line fix a unique plane.

Teacher's Note:
1. Think of a tripod: three legs fix the ground plane.
2. Non-collinearity is needed.

Answer: (b) \( 169 \)

Explanation:
1. \( 12^2+5^2=144+25=169 \).
2. This equals \( AC^2=13^2 \), so the triangle is right-angled at \( B \).

Teacher's Note:
1. Compare the sum of the smaller squares with the largest square.
2. (12, 5, 13) is a Pythagorean triple.

Answer: (c) intersect

Explanation:
1. Suppose \( l \) does not meet \( n \); then \( l\parallel n \).
2. Together with \( n\parallel m \), this gives \( l\parallel m \), contradicting that \( l \) meets \( m \).
3. So \( l \) intersects \( n \).

Teacher's Note:
1. Proof by contradiction.
2. Parallelism is transitive.