Maharashtra Board Class 9 Maths Part 2 Geometry Chapter 1 Basic Concepts in Geometry Set 1.2 Solutions

Question 1. The following table shows points on a number line and their co-ordinates. Decide whether the pair of segments given below the table are congruent or not.
ℹ️ चित्र व्याख्या (Diagram Explanation): एक तालिका दी गई है जो एक संख्या रेखा पर बिंदुओं A, B, C, D, E और उनके संगत निर्देशांकों को दर्शाती है। बिंदु A का निर्देशांक -3, B का 5, C का 2, D का -7, और E का 9 है।

(i) seg DE and seg AB
(ii) seg BC and seg AD
(iii) seg BE and seg AD
Answer: Solution:
(i) Co-ordinate of the point E is 9. Co-ordinate of the point D is -7. Since, 9 > -7 \( \therefore \) d(D, E) = 9 - (-7) = 9 + 7 = 16 \( \therefore \) l(DE) = 16 ...(i) Co-ordinate of the point A is -3. Co-ordinate of the point B is 5. Since, 5 > -3 \( \therefore \) d(A, B) = 5 - (-3) = 5 + 3 = 8 \( \therefore \) l(AB) = 8 ...(ii) \( \therefore \) l(DE) \( \ne \) l(AB) ...[From (i) and (ii)] \( \therefore \) seg DE and seg AB are not congruent.
(ii) Co-ordinate of the point B is 5. Co-ordinate of the point C is 2. Since, 5 > 2 \( \therefore \) d(B, C) = 5 - 2 = 3 \( \therefore \) l(BC) = 3 ...(i) Co-ordinate of the point A is -3. Co-ordinate of the point D is -7. Since, -3 > -7 \( \therefore \) d(A, D) = -3 - (-7) = -3 + 7 = 4 \( \therefore \) l(AD) = 4 . ...(ii) \( \therefore \) l(BC) \( \ne \) l(AD) ... [From (i) and (ii)] \( \therefore \) seg BC and seg AD are not congruent.
(iii) Co-ordinate of the point E is 9. Co-ordinate of the point B is 5. Since, 9 > 5 \( \therefore \) d(B, E) = 9 - 5 = 4 \( \therefore \) l(BE) = 4 ...(i) Co-ordinate of the point A is -3. Co-ordinate of the point D is -7. Since, -3 > -7 \( \therefore \) d(A, D) = -3 - (-7) = 4 \( \therefore \) l(AD) = 4 ...(ii) \( \therefore \) l(BE) = l(AD) ...[From (i) and (ii)] \( \therefore \) seg BE and seg AD are congruent. i.e, seg BE \( \cong \) seg AD In simple words: To check if segments are congruent, calculate their lengths using the absolute difference of their coordinates. If the lengths are equal, the segments are congruent; otherwise, they are not. This method is applied to three pairs of segments.

🎯 Exam Tip: Remember that two segments are congruent if and only if their lengths are equal. Always use the distance formula \( d(X, Y) = |x - y| \) where x and y are the coordinates of points X and Y respectively.

Question 2. Point M is the midpoint of seg AB. If AB = 8, then find the length of AM.
Answer: Solution: Point M is the midpoint of seg AB and l(AB) = 8. ...[Given]
ℹ️ चित्र व्याख्या (Diagram Explanation): एक रेखाखंड AB दिखाया गया है, जिस पर बिंदु M स्थित है। बिंदु M, रेखाखंड AB का मध्यबिंदु है, जिसका अर्थ है कि यह AB को दो बराबर भागों में विभाजित करता है। \[ I(AM) = \frac{1}{2} I(AB) \] ...[ \( \therefore \) M is midpoint of seg AB]
\( \therefore \) \[ I(AM) = \frac{1}{2} \times 8 = 4 \] \[ I(AM) = 4 \] In simple words: If M is the midpoint of segment AB, then the length of AM is half the length of AB. Given AB = 8, AM is 4.

🎯 Exam Tip: The midpoint of a segment divides it into two segments of equal length. This relationship is crucial for calculating segment lengths when a midpoint is given.

Question 3. Point P is the midpoint of seg CD. If CP = 2.5, find I(CD).
Answer: Solution: Point P is the midpoint of seg CD and l(CP) = 2.5 ...[Given]
ℹ️ चित्र व्याख्या (Diagram Explanation): एक रेखाखंड CD दिखाया गया है, जिस पर बिंदु P स्थित है। बिंदु P, रेखाखंड CD का मध्यबिंदु है, जिसका अर्थ है कि यह CD को दो बराबर भागों में विभाजित करता है। \[ I(CP) = \frac{1}{2} I(CD) \] ...[ \( \therefore \) P is midpoint of seg CD]
\( \therefore \) \[ 2.5 = \frac{1}{2} \times I(CD) \] \( \therefore \) l(CD) = 2.5 x 2 \( \therefore \) l(CD) = 5 In simple words: Since P is the midpoint of segment CD, the length of CD is twice the length of CP. Given CP = 2.5, CD is 5.

🎯 Exam Tip: When a midpoint is given, you can find the total length of the segment by doubling the length of one of the halves. This is a common application of midpoint properties.

Question 4. If AB = 5 cm, BP = 2 cm and AP = 3.4 cm, compare the segments.
Answer: Solution: Given, l(AB) = 5 cm, l(BP) = 2 cm, l(AP) = 3.4 cm ...[Given] Since, 2 < 3.4 < 5
\( \therefore \) l(BP) < l(AP) < l(AB) i.e., seg BP < seg AP < seg AB In simple words: To compare segments, simply compare their given lengths. Arrange the lengths in ascending or descending order to determine the relative sizes of the segments.

🎯 Exam Tip: Comparing segments involves ordering their lengths. The segment with the smaller length is considered "smaller" than the segment with the larger length.

Question 5. Write the answers to the following questions with reference to the figure given below:
ℹ️ चित्र व्याख्या (Diagram Explanation): एक संख्या रेखा दिखाई गई है जिस पर बिंदु T, S, R, P, Q क्रम से स्थित हैं। यह रेखाखंड और किरणें बनाने के लिए एक संदर्भ प्रदान करती है।
(i) Write the name of the opposite ray of ray RP
(ii) Write the intersection set of ray PQ and ray RP.
(iii) Write the union set of ray PQ and ray QR.
(iv) State the rays of which seg QR is a subset.
(v) Write the pair of opposite rays with common end point R.
(vi) Write any two rays with common end point S.
(vii) Write the intersection set of ray SP and ray ST.
Answer:
(i) Ray RS or ray RT
(ii) Ray PQ
(iii) Line QR
(iv) Ray QR, ray QS, ray QT, ray RQ, ray SQ, ray TQ
(v) Ray RP and ray RS, ray RQ and ray RT vi. Ray ST, ray SR
(vii) Point S In simple words: This question tests understanding of basic geometry concepts like rays, opposite rays, intersection, and union of rays, and subsets of rays, all relative to points on a line.

🎯 Exam Tip: Clearly identify the direction and starting point of each ray. Opposite rays share a common endpoint and extend in opposite directions. The intersection of two rays can be a ray, a segment, or a point, while the union can form a line or a larger ray.

Question 6. Answer the questions with the help of figure given below.
ℹ️ चित्र व्याख्या (Diagram Explanation): एक संख्या रेखा दी गई है जिस पर बिंदु R, U, Q, L, P, A, C, V, B, D स्थित हैं, और उनके संगत निर्देशांक दिखाए गए हैं: R(-6), U(-4), Q(-2), L(0), P(2), A(4), C(4), V(5), B(6), D(6).
(i) State the points which are equidistant from point B.
(ii) Write a pair of points equidistant from point Q.
(iii) Find d(U,V), d(P,C), d(V,B), d(U, L).
Answer:
(i) Points equidistant from point B are a. A and C, because d(B, A) = d(B, C) = 2 b. D and P, because d(B, D) = d(B, P) = 4
(ii) Points equidistant from point Q are a. L and U, because d(Q, L) = d(Q, U) = 1 b. P and R, because d(P, Q) = d(Q, R) = 2
(iii) a. Co-ordinate of the point U is -5. Co-ordinate of the point V is 5. Since, 5 > -5 \( \therefore \) d(U, V) = 5 - (-5) = 5 + 5 \( \therefore \) d(U, V) = 10
b. Co-ordinate of the point P is -2. Co-ordinate of the point C is 4. Since, 4 > -2 \( \therefore \) d(P, C) = 4 - (-2) = 4 + 2 \( \therefore \) d(P, C) = 6
c. Co-ordinate of the point V is 5. Co-ordinate of the point B is 2. Since, 5 > 2 \( \therefore \) d(V, B) = 5 - 2 \( \therefore \) d(V, B) = 3
d. Co-ordinate of the point U is -5. Co-ordinate of the point L is -3. Since, -3 > -5 \( \therefore \) d(U, L) = -3 - (-5) = -3 + 5 \( \therefore \) d(U, L) = 2 In simple words: This question involves identifying points equidistant from a given point on a number line by calculating distances, and also calculating the distance between various pairs of points using their coordinates.

🎯 Exam Tip: To find points equidistant from a given point, calculate the distance from the given point to all other points and compare them. Remember the distance formula \( d(X, Y) = |x - y| \) for points on a number line.