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Piyush365
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Let the numbers be x and y with x > y. The equations are x – y = 26 and x = 3y. Substituting gives 3y – y = 26, so 2y = 26, y = 13 and x = ...

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Piyush365
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Let the integers be x and y. Then x + y = 5 and x – y = -21. Adding both equations gives 2x = -16, so x = -8. Substituting yields y = 13. The numbers are -8 and ...

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Piyush365
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The shaded fraction is 1/5. The parallel lines connecting vertices and opposite midpoints are perpendicular, dividing the square into five congruent smaller squares by geometric dissection, making the central shaded square exactly one-fifth of the total area. Class 9 Maths Ganita ...

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Piyush365
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Method 1: 180° edge rotations place all four interior angles around each shared vertex, summing to 360° without gaps or overlaps. Methods 2 and 3: Parallelograms tile the plane, so positioning copies preserves edge congruences and seamlessly tiles the entire ...

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Piyush365
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Multiple converses to a theorem. Let us see how multiple statements can be considered converses to the Midpoint Theorem and how Theorem 7 is one of them. To formulate a converse we should express the original statement in “If … then …” form. For a complex statement, there may be multiple ways to do that. The Midpoint Theorem starts with △ABC and points P and Q on sides AB and AC respectively. The theorem has two assumptions and two conclusions, which we have named for further discussion. Assumptions: (P MID) P is the midpoint of AB and (Q MID) Q is the midpoint of AC. Conclusions: (PRLL) PQ ∥ BC and (HALF) PQ = BC/2. Let us use these four named conditions to discuss various possible statements. (i) A natural converse of the Midpoint Theorem would be: “If (PRLL) and (HALF) are true, then (P MID) and (Q MID) are true.” Write out this statement fully. Experiment and see that it seems to be true! Prove the statement. (Hint: Try to run a proof of the Midpoint Theorem backwards.) (ii) To see Theorem 7 as a converse of the Midpoint Theorem, we first write the Midpoint Theorem as follows: “Suppose P is the midpoint of side AB of △ABC and Q is a point on side AC. If Q is the midpoint of AC then PQ ∥ BC.” (Also PQ = BC/2, but let us set that aside for now.) Write the converse of the second sentence in quotes by keeping the first sentence the same. Verify that Theorem 7 is what you get! (And PQ = BC/2 also follows.) Now write Theorem 7 in terms of the four named conditions. (iii) The discussion so far suggests that it is reasonable to take two of the four listed statements as assumptions and ask if the other two are true. Verify that only one such combination remains to be examined. “If (P MID) and (HALF) are true, then can we conclude (Q MID) and/or (PRLL)?” Write this out in words. This is a precise version of question (1) stated before Theorem 7. Can you answer it?

(i) If a segment joining two sides is parallel to the third side and half its length, it connects their midpoints. (ii) “If PQ ∥ BC, then Q is the midpoint of AC”; represented as: If (P MID) and (PRLL), ...

(q mid) q is the midpoint of aca natural converse of the midpoint theoremclass 9 maths ganita manjari part 2 chapter 12 question answerganita manjari part 2 chapter 12 quadrilaterals solutionsif (p mid) and (half) are truemidpoint theorem starts with △abc and points p and qsuppose p is the midpoint of side ab of △abc and q
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