CBSE Class 6 Maths Chapter 2 Geometry NCERT Solutions

To find the number of lines that can be drawn through a set of points where no three points are collinear, we need to choose any two points to form a line. This is a combination problem. The formula for combinations is given by C(n, k) = \(\frac{n!}{k!(n-k)!}\), where 'n' is the total number of items, and 'k' is the number of items to choose. In this case, we have 5 points (n=5) and we need to choose 2 points to form a line (k=2).

Number of lines = C(5, 2) = \(\frac{5!}{2!(5-2)!}\) = \(\frac{5!}{2!3!}\) = \(\frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)}\)

Number of lines = \(\frac{5 \times 4}{2 \times 1}\) = \(\frac{20}{2}\) = 10.

Therefore, 10 lines can pass through five points if no three of them are collinear. Option (A) is correct.

A septagon is a polygon with 7 sides. The formula to calculate the number of diagonals in a polygon with 'n' sides is given by:

Number of diagonals = \(\frac{n(n-3)}{2}\)

For a septagon, n = 7.

Substituting n = 7 into the formula:

Number of diagonals = \(\frac{7(7-3)}{2}\) = \(\frac{7(4)}{2}\) = \(\frac{28}{2}\) = 14.

So, the number of diagonals in a septagon is 14. Option (D) is correct.

To find the number of line segments in the given figure, we need to identify all possible pairs of points that form a line segment. Let's list them systematically. The points are A, B, C, D, and E. We can assume these points are arranged in a way that forms a pentagon or a similar structure where all pairs form distinct line segments.

The line segments are formed by connecting any two points. We can list them as follows:

• Starting from A: AB, AC, AD, AE (4 segments)
• Starting from B (excluding AB which is already counted): BC, BD, BE (3 segments)
• Starting from C (excluding AC, BC): CD, CE (2 segments)
• Starting from D (excluding AD, BD, CD): DE (1 segment)

Total number of line segments = 4 + 3 + 2 + 1 = 10.

Alternatively, using the combination formula C(n, 2) where n is the number of points (n=5):

Number of line segments = C(5, 2) = \(\frac{5 \times (5-1)}{2}\) = \(\frac{5 \times 4}{2}\) = \(\frac{20}{2}\) = 10.

Therefore, there are 10 line segments in the given figure. Option (B) is correct.

At 9 o'clock, the minute hand of the clock points exactly at the number 12. The hour hand points exactly at the number 9.

A clock face is a circle, which has 360 degrees. There are 12 numbers on the clock face, so the angle between any two consecutive numbers is \(\frac{360^{\circ}}{12}\) = 30 degrees.

The numbers between 9 and 12 on the clock face are 10, 11, and 12. There are 3 intervals between the hour hand (at 9) and the minute hand (at 12).

The angle between the hour hand and the minute hand is the number of intervals multiplied by the degrees per interval:

Angle = 3 intervals \(\times\) 30 degrees/interval = 90 degrees.

This is the smaller angle between the hands. The other angle (the reflex angle) is the total degrees in a circle minus the smaller angle:

Reflex Angle = 360 degrees - 90 degrees = 270 degrees.

So, the two angles between the hour and minute hands at 9 o'clock are 90° and 270°. Option (B) is correct.

A bicycle wheel is circular, and the spokes radiate from the center to the rim. The total angle around the center of the wheel is 360 degrees.

If there are 48 spokes, these spokes divide the circle into 48 equal sections. To find the angle between any two consecutive spokes, we need to divide the total degrees in a circle by the number of spokes.

Angle between consecutive spokes = \(\frac{\text{Total degrees in a circle}}{\text{Number of spokes}}\)

Angle = \(\frac{360^{\circ}}{48}\)

To simplify the fraction \(\frac{360}{48}\):

We can divide both numerator and denominator by common factors. For example, both are divisible by 12:

\(\frac{360 \div 12}{48 \div 12}\) = \(\frac{30}{4}\)

Now, \(\frac{30}{4}\) = 7.5.

So, the angle between a pair of two consecutive spokes is 7.5 degrees. Option (B) is correct.

The notation ∠XYZ refers to the angle formed at vertex Y, with rays YX and YZ. This angle can also be simply denoted as ∠Y, as Y is the vertex.

Let's analyze the given options:

• (A) ∠Y: This notation correctly represents the angle at vertex Y, which is ∠XYZ.
• (B) ∠ZXY: This notation represents the angle at vertex X, formed by rays XZ and XY. This is different from ∠XYZ.
• (C) ∠ZYX: This notation also represents the angle at vertex Y, formed by rays YZ and YX. It is the same as ∠XYZ.
• (D) ∠XYP: Assuming P is a point on the ray extending from X through Y, this notation might represent an angle related to X, Y, and P. However, based on standard geometric notation where the middle letter is the vertex, if P is a point such that X, Y, P are collinear in that order, then ∠XYP would be 180°. If P is on the ray PX, and Y is another point, ∠XYP would be the angle at Y. Without a clear diagram showing P's exact position relative to X and Y, we rely on the most definitive incorrect option.

The question asks which notation *cannot* be used to write ∠XYZ. ∠ZXY clearly denotes the angle at vertex X, not vertex Y. Therefore, ∠XYZ cannot be written as ∠ZXY.

Option (B) is correct.

The problem describes a scenario involving points and angles on a ray. Let's assume 'a' is a point, and it is shifted to point B along the ray PX. This implies that P, A, and B are points on the ray PX, likely in that order or with A between P and B.

The condition given is PB = 2PA. This means the distance from P to B is twice the distance from P to A.

Let's consider the ray PX and another ray PY originating from P, forming an angle ∠XPY.

If point A is on the ray PX, and point B is also on the ray PX such that PB = 2PA, this implies that A is closer to P than B is. For example, if PA = 5 units, then PB = 10 units. This means A is between P and B.

The question asks for the measure of ∠BPY. However, the description seems to imply that point A is being shifted *to* point B, and the condition PB = 2PA relates their distances from P. The angle ∠BPY depends on the position of ray PY relative to ray PX.

If we interpret 'point a' as point A, and it is shifted to point B along the ray PX, it means A and B are on the ray PX. The condition PB = 2PA means B is further from P than A is. The angle ∠BPY is the angle between the ray PB (which is the same as ray PX since B is on PX) and the ray PY.

Therefore, ∠BPY is the same as ∠XPY.

The problem statement is slightly ambiguous as it mentions shifting 'point a' to 'point B' and then asks for ∠BPY. If A and B are on the ray PX, then the ray PB is the same as the ray PX. So ∠BPY is the same as ∠XPY.

Without additional information or a diagram (Fig 2.7), it's difficult to definitively determine the angle ∠BPY. However, if we assume that the original point was A, and it moved to B along PX, and the question is about the angle formed by the new position (B) and ray PY, then ∠BPY = ∠XPY.

If we assume that the question implies a relationship between the angle and the distances, and perhaps there was an initial angle related to A, like ∠APY, and now we are considering ∠BPY. If A and B are on ray PX, then ∠BPY = ∠XPY.

Let's reconsider the phrasing: "if point a is shifted to point B along the ray PX such that PB = 2PA". This suggests A and B are points on the ray PX. The angle ∠BPY is the angle between the ray PB (which is the same as ray PX) and ray PY. So, ∠BPY = ∠XPY.

The options provided are related to 45° and 90°. This suggests there might be a standard geometric configuration implied or missing information.

If we assume that the original setup involved a point A such that ∠APY was some value, and now point B is further along PX, and we are asked about ∠BPY. Since B is on the ray PX, the ray PB is identical to the ray PX. Thus, ∠BPY is the same angle as ∠XPY.

The problem might be testing the understanding that if B lies on the ray PX, then the angle ∠BPY is simply the angle between ray PX and ray PY.

Given the options, and the fact that the source material is for Class 6, it's likely a conceptual question. If B is on ray PX, then ray PB is the same as ray PX. Therefore, ∠BPY is the angle formed by ray PX and ray PY.

Without Fig 2.7, we cannot determine the exact angle. However, if we assume a standard context where such questions appear, and given the options, there might be an implicit assumption about the angle.

Let's assume the question implies that the angle is fixed regardless of the position of B on the ray PX, as long as B is on PX. Then ∠BPY = ∠XPY.

If we consider a scenario where A was initially at some position, and B is further along PX, and the angle is being measured. The condition PB = 2PA relates distances, not angles directly, unless there's a proportionality involved which is not stated.

Let's consider the possibility that the question is flawed or relies heavily on the missing figure. However, if we must choose from the options, and knowing that B is on ray PX, then ∠BPY is the angle between ray PX and ray PY.

If we assume that the original point was A, and it was shifted to B along PX, and the angle ∠APY was, for example, 45 degrees, and now we are considering ∠BPY. Since B is on PX, ∠BPY = ∠XPY. If the angle ∠XPY itself is 45 degrees, then option (B) would be correct.

Let's assume the question is asking about the angle formed by the ray PY and the ray PX (since B is on PX). If the angle ∠XPY were, for instance, 45 degrees, then ∠BPY would also be 45 degrees. The condition PB = 2PA might be a distractor or relevant if A and B were on a different ray, or if there was a relationship between angles and distances.

Given the options, and the common angles used in geometry problems, 45° is a frequent value. If we assume that the angle ∠XPY is 45°, then ∠BPY would also be 45° because B lies on the ray PX.

Let's assume the intended meaning is that the angle formed by ray PY with the ray PX is a specific value. Since B is on ray PX, the ray PB is the same as ray PX. Therefore, ∠BPY = ∠XPY. If the angle ∠XPY is 45°, then ∠BPY is 45°.

Option (B) is the most plausible answer if we assume ∠XPY = 45°.