Class 9 Maths

Piyush365

Asked: October 5, 2026In: Class 9 Maths

Complete the following proof of the Midpoint Theorem. Extend segment PQ beyond Q until point S. By how much should we extend PQ? It would be good to be able to prove that △APQ ≅ △CSQ. (Why?) Use this as a guide to specify the location of S and then complete this proof. (ii) Give a similar proof of the converse of the Midpoint Theorem (Theorem 7). Start by extending PQ up to a suitable point S.

Extend PQ so QS = PQ. By SAS, △APQ ≅ △CSQ, making CS = AP = PB and CS ∥ PB. Thus PBCS is a parallelogram, giving PQ ∥ BC and PQ = BC/2. For (ii), complete PBCS similarly to ...

Piyush365

Asked: October 5, 2026In: Class 9 Maths

Let P, Q, R, S be four points on sides AB, BC, CD and DA respectively of a quadrilateral ABCD. Suppose PQRS is a parallelogram. Must P, Q, R and S be midpoints of the respective sides? This can be considered a possible converse question to Theorem 9.

No, they do not have to be midpoints. In a parallelogram ABCD, choosing points dividing opposite sides in equal ratios (like AP = CR = 1/3 AB and BQ = DS = 1/3 BC) creates an inscribed parallelogram that is ...

Piyush365

Asked: October 5, 2026In: Class 9 Maths

Is the Midpoint Theorem for Quadrilaterals (Theorem 9) true when the quadrilateral is non-convex? How about when it is self-intersecting? Experiment and check. The pictures may look strange, but the theorem appears to be true. (i) Can you explain why this is so? See the geometric reasoning in the text. (ii) There is one exception: In a very special case the Varignon parallelogram becomes a single segment. When will this happen? (iii) Does your reasoning apply even when ABCD is non-planar?

Yes, it holds in both cases. (i) The Midpoint Theorem on triangles formed by diagonals still ensures opposite sides are parallel and equal to half the diagonal. (ii) It degenerates into a segment when diagonal AC is parallel to BD ...

Piyush365

Asked: October 5, 2026In: Class 9 Maths

(i) Suppose ABCD is a parallelogram and M, N are midpoints of AB and CD respectively. Show that segments DM and BN trisect segment AC. (ii) Use part (i) to find a procedure to trisect any given segment PQ. Find two other ways to trisect PQ, one using Exercise 11 above and a third using the Centroid Theorem.

(i) MBND is a parallelogram, so MD ∥ BN. Applying Theorem 7 to △ABY and △CDX proves the intercepts along AC are equal. (ii) Construct a parallelogram with diagonal PQ, draw parallel lines dividing an auxiliary ray into three parts ...

Piyush365

Asked: October 5, 2026In: Class 9 Maths

A farm has some cows, sheep and chickens. Last year the cows made up 60%, the sheep 25% and the chickens 15%. There was a decrease in the number of all three animals’ population over the year. Choose the possibilities for the change in their respective shares of the population— (i) % of cows decreased, % of sheep decreased, % of chickens decreased (ii) % of cows increased, % of sheep increased, % of chickens increased (iii) % of cows remained the same, % of sheep remained the same, % of chickens remained the same (iv) % of cows decreased, % of sheep increased, % of chickens remained the same (v) % of cows increased, % of sheep increased, % of chickens decreased.

Possibilities (iii), (iv) and (v) are possible. The sum of percentage shares must always equal 100%, so all shares cannot simultaneously increase or simultaneously decrease together, ruling out (i) and (ii). Class 9 Ganita Manjari Part 2 Chapter 10 Solutions Class ...

Category: Class 9 Maths

Let P, Q, R, S be four points on sides AB, BC, CD and DA respectively of a quadrilateral ABCD. Suppose PQRS is a parallelogram. Must P, Q, R and S be midpoints of the respective sides? This can be considered a possible converse question to Theorem 9.

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