Maharashtra Board Class 9 Maths Part 2 Geometry Chapter 3 Set 3.5 Triangles Solutions

Question 1. If ΔXYZ ~ ΔLMN, write the corresponding angles of the two triangles and also write the ratios of corresponding sides.
Solution:
ΔΧΥΖ ~ ΔLMN [Given]
∴∠X ≃ ∠L
∠Y ≃ ∠M
∠Z = ∠N [Corresponding angles of similar triangles]
\(\frac{XY}{LM} = \frac{YZ}{MN} = \frac{XZ}{PN}\) [Corresponding sides of similar triangles]
In simple words: When two triangles are similar, their corresponding angles are congruent, and the ratios of their corresponding sides are equal.

🎯 Exam Tip: Remember to correctly match the vertices when writing corresponding angles and sides for similar triangles to avoid errors.

Question 2. In ΔΧΥΖ, XY = 4 cm, YZ = 6 cm, XZ = 5 cm. If ∆XYZ ~ ΔPQR and PQ = 8 cm, then find the lengths of remaining sides of ΔPQR.
Solution:
ΔΧΥΖ ~ ΔPQR [Given]
∴ \(\frac{XY}{PQ} = \frac{YZ}{QR} = \frac{XZ}{PR}\) [Corresponding sides of similar triangles]
∴ \(\frac{4}{8} = \frac{6}{QR} = \frac{5}{PR}\) (i)
Now, \(\frac{4}{8} = \frac{6}{QR}\)
QR = \(\frac{6 \times 8}{4}\)
QR = 12 cm
ii. Also, \(\frac{4}{8} = \frac{5}{PR}\) [From (i)]
PR = \(\frac{5 \times 8}{4}\)
PR = 10 cm
∴ QR = 12 cm, PR = 10cm
In simple words: Given similar triangles and the lengths of one triangle's sides, along with one corresponding side of the second triangle, we can find the remaining sides by using the constant ratio of corresponding sides.

🎯 Exam Tip: Clearly write down the ratio of corresponding sides. This step is crucial for accurate calculation of unknown lengths in similar figures.

Question 3. Draw a sketch of a pair of similar triangles. Label them. Show their corresponding angles by the same signs. Show the lengths of corresponding sides by numbers in proportion.
Solution:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो समरूप त्रिभुजों, ΔGHI और ΔSTU को दर्शाता है। ΔGHI की भुजाओं की लंबाई 1, 3, 2 के रूप में चिह्नित हैं, जबकि ΔSTU की संगत भुजाओं की लंबाई 2, 6, 4 के रूप में चिह्नित हैं, जो 1:2 का अनुपात बनाए रखती हैं। उनके संगत कोणों को समान संकेतकों (positional correspondence) से दर्शाया गया है, जिसमें G का संगत S, H का T और I का U है।
ΔGHI ~ ΔSTU
In simple words: This diagram illustrates two triangles, ΔGHI and ΔSTU, that are similar, meaning their corresponding angles are equal and their corresponding sides are in a constant proportion.

🎯 Exam Tip: When drawing similar triangles, ensure corresponding angles are marked with the same symbols and corresponding sides maintain a consistent ratio for full marks.

Maharashtra Board Class 9 Maths Chapter 3 Triangles Practice Set 3.5 Intext Questions And Activities

Question 1. We have learnt that if two triangles are equiangular then their sides are in proportion. What do you think if two quadrilaterals are equiangular? Are their sides in proportion? Draw different figures and verify. Verify the same for other polygons. (Textbook pg no 50)
Answer:
If two quadrilaterals are equiangular then their sides will not necessarily be in proportion.
Case 1: The two quadrilaterals are of the same type.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वर्ग ABCD और PQRS को दर्शाता है। वर्ग ABCD के शीर्ष दक्षिणावर्त क्रम में A, B, C, D हैं। वर्ग PQRS के शीर्ष दक्षिणावर्त क्रम में P, Q, R, S हैं। दोनों समकोणिक हैं (सभी कोण 90 डिग्री हैं)।
Consider squares ABCD and PQRS.
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R, ∠D = ∠S
\(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CD}{RS} = \frac{AD}{PS}\)
Case 2: The two quadrilaterals are of different types.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक वर्ग ABCD और एक आयत STUV को दर्शाता है। वर्ग ABCD के शीर्ष A, B, C, D हैं। आयत STUV के शीर्ष S, T, U, V हैं। दोनों समकोणिक हैं (सभी कोण 90 डिग्री हैं)।
Consider square ABCD and rectangle STUV.
∠A = ∠S, ∠B = ∠T, ∠C = ∠U, ∠D = ∠V
Now, \(\frac{AB}{ST} = \frac{CD}{UV}\) and \(\frac{BC}{TU} = \frac{AD}{SV}\)
But \(\frac{AB}{ST} + \frac{BC}{TU}\)
In simple words: While equiangular triangles always have proportional sides, equiangular quadrilaterals or other polygons do not necessarily have proportional sides unless they are of the same specific type (e.g., two squares).

🎯 Exam Tip: This question highlights a key difference between similarity in triangles and other polygons. For polygons with more than three sides, both equiangularity and proportionality of sides are required for similarity, not just one of them.