By the Midpoint Theorem in △ABC, the segment MN joining the midpoints of AB and AC is parallel to base BC. Consequently, the line MN is parallel to segment BD on BC. Now consider △ABD. The line containing MN passes through the midpoint M of AB and is parallel to BD. By the Converse of the Midpoint TRead more
By the Midpoint Theorem in △ABC, the segment MN joining the midpoints of AB and AC is parallel to base BC. Consequently, the line MN is parallel to segment BD on BC. Now consider △ABD. The line containing MN passes through the midpoint M of AB and is parallel to BD. By the Converse of the Midpoint Theorem, it bisects the third side AD.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
Let K be the midpoint of AD. In △DAB, KH joins midpoints, so KH ∥ AB. In △ADC, KG joins midpoints, so KG ∥ DC ∥ AB. By Euclid's parallel postulate, points K, H, G lie on the same straight line, so line GH ∥ AB. We assume AB ≠ CD because if AB = CD, ABCD is a parallelogram whose diagonals bisect eachRead more
Let K be the midpoint of AD. In △DAB, KH joins midpoints, so KH ∥ AB. In △ADC, KG joins midpoints, so KG ∥ DC ∥ AB. By Euclid’s parallel postulate, points K, H, G lie on the same straight line, so line GH ∥ AB. We assume AB ≠ CD because if AB = CD, ABCD is a parallelogram whose diagonals bisect each other, making G and H the exact same point.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
(i) By the Midpoint Theorem, PQ ∥ AC ∥ SR and PS ∥ BD ∥ QR, meaning PQRS is a parallelogram whose diagonals PR and QS bisect each other. (ii) Since PQ = AC/2 and QR = BD/2, if AC = BD, all sides of PQRS are equal, making it a rhombus whose diagonals are perpendicular. The converse is true: perpendicRead more
(i) By the Midpoint Theorem, PQ ∥ AC ∥ SR and PS ∥ BD ∥ QR, meaning PQRS is a parallelogram whose diagonals PR and QS bisect each other.
(ii) Since PQ = AC/2 and QR = BD/2, if AC = BD, all sides of PQRS are equal, making it a rhombus whose diagonals are perpendicular. The converse is true: perpendicular diagonals in parallelogram PQRS make it a rhombus, which forces AC = BD.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
(i) Reconstruct △A'PS congruent to △APS. Extend A'P past P by length A'P to locate B', extend B'Q through Q by length B'Q to find C' and extend C'R through R by length C'R to find D'. (ii) Successive point reflections across midpoints P, Q, R, S guarantee each segment is straight and preserved in leRead more
(i) Reconstruct △A’PS congruent to △APS. Extend A’P past P by length A’P to locate B’, extend B’Q through Q by length B’Q to find C’ and extend C’R through R by length C’R to find D’.
(ii) Successive point reflections across midpoints P, Q, R, S guarantee each segment is straight and preserved in length, yielding △SDR ≅ △SD’R and making A’B’C’D’ ≅ ABCD.
(iii) In PQRS, sides are parallel to diagonals AC, BD and equal half their lengths. If PQRS is a square, its sides are equal and perpendicular, forcing AC = BD and AC ⟂ BD. Conversely, perpendicular and equal diagonals create four equal sides with right angles, forming a square.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
Draw non-convex quadrilateral DART with an internal dent angle exceeding 180°. In Method 1, repeatedly rotate copies by 180° around the midpoints of each shared edge, creating an interlocking plane tiling. In Method 2, build a grid using the Varignon parallelogram of DART and position copies onto alRead more
Draw non-convex quadrilateral DART with an internal dent angle exceeding 180°. In Method 1, repeatedly rotate copies by 180° around the midpoints of each shared edge, creating an interlocking plane tiling. In Method 2, build a grid using the Varignon parallelogram of DART and position copies onto alternating cells. Method 1 is generally preferred for its simplicity and directness without pre-drawing a grid.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
In △ABC, let M and N be midpoints of AB and AC respectively. Let D be any point on BC. Show that MN bisects AD.
By the Midpoint Theorem in △ABC, the segment MN joining the midpoints of AB and AC is parallel to base BC. Consequently, the line MN is parallel to segment BD on BC. Now consider △ABD. The line containing MN passes through the midpoint M of AB and is parallel to BD. By the Converse of the Midpoint TRead more
By the Midpoint Theorem in △ABC, the segment MN joining the midpoints of AB and AC is parallel to base BC. Consequently, the line MN is parallel to segment BD on BC. Now consider △ABD. The line containing MN passes through the midpoint M of AB and is parallel to BD. By the Converse of the Midpoint Theorem, it bisects the third side AD.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-12/
See lessIn a quadrilateral ABCD, suppose AB ∥ DC and AB ≠ CD. Suppose G and H are the midpoints of AC and BD respectively. Prove that GH ∥ AB. (Why did we assume AB ≠ CD?)
Let K be the midpoint of AD. In △DAB, KH joins midpoints, so KH ∥ AB. In △ADC, KG joins midpoints, so KG ∥ DC ∥ AB. By Euclid's parallel postulate, points K, H, G lie on the same straight line, so line GH ∥ AB. We assume AB ≠ CD because if AB = CD, ABCD is a parallelogram whose diagonals bisect eachRead more
Let K be the midpoint of AD. In △DAB, KH joins midpoints, so KH ∥ AB. In △ADC, KG joins midpoints, so KG ∥ DC ∥ AB. By Euclid’s parallel postulate, points K, H, G lie on the same straight line, so line GH ∥ AB. We assume AB ≠ CD because if AB = CD, ABCD is a parallelogram whose diagonals bisect each other, making G and H the exact same point.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-12/
See lessSuppose the midpoints of sides AB, BC, CD and DA of a quadrilateral ABCD are P, Q, R and S respectively. (i) Show that PR and QS bisect each other. (ii) Show that if AC = BD, then PR and QS are perpendicular. Is the converse true?
(i) By the Midpoint Theorem, PQ ∥ AC ∥ SR and PS ∥ BD ∥ QR, meaning PQRS is a parallelogram whose diagonals PR and QS bisect each other. (ii) Since PQ = AC/2 and QR = BD/2, if AC = BD, all sides of PQRS are equal, making it a rhombus whose diagonals are perpendicular. The converse is true: perpendicRead more
(i) By the Midpoint Theorem, PQ ∥ AC ∥ SR and PS ∥ BD ∥ QR, meaning PQRS is a parallelogram whose diagonals PR and QS bisect each other.
(ii) Since PQ = AC/2 and QR = BD/2, if AC = BD, all sides of PQRS are equal, making it a rhombus whose diagonals are perpendicular. The converse is true: perpendicular diagonals in parallelogram PQRS make it a rhombus, which forces AC = BD.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-12/
See lessSuppose PQRS is the Varignon parallelogram of ABCD. (i) Copy only PQRS on another paper. Show how you will recreate a congruent copy A’B’C’D’ of ABCD from PQRS. This will be relevant when we return to study tilings at the end of this chapter. (Hint: How will you place vertex A’? How will you place B’, C’ and D’?) (ii) There are multiple ways to construct A’B’C’D’ in (i). Justify why the quadrilateral A’B’C’D’ you constructed is congruent to ABCD. You may need to show why S is collinear with the points A’ and D’ that you constructed and similarly for P, Q and R. (Hint: Use congruence of triangles, for example △SDR ≅ △SD’R.) (iii) Show that if PQRS is a square, then AC and BD are perpendicular and equal. Prove the converse.
(i) Reconstruct △A'PS congruent to △APS. Extend A'P past P by length A'P to locate B', extend B'Q through Q by length B'Q to find C' and extend C'R through R by length C'R to find D'. (ii) Successive point reflections across midpoints P, Q, R, S guarantee each segment is straight and preserved in leRead more
(i) Reconstruct △A’PS congruent to △APS. Extend A’P past P by length A’P to locate B’, extend B’Q through Q by length B’Q to find C’ and extend C’R through R by length C’R to find D’.
(ii) Successive point reflections across midpoints P, Q, R, S guarantee each segment is straight and preserved in length, yielding △SDR ≅ △SD’R and making A’B’C’D’ ≅ ABCD.
(iii) In PQRS, sides are parallel to diagonals AC, BD and equal half their lengths. If PQRS is a square, its sides are equal and perpendicular, forcing AC = BD and AC ⟂ BD. Conversely, perpendicular and equal diagonals create four equal sides with right angles, forming a square.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-12/
See lessDraw a non-convex 4-gon DART. Show how we can tile the plane with copies of DART. Both methods that we discussed earlier will work. Which do you prefer?
Draw non-convex quadrilateral DART with an internal dent angle exceeding 180°. In Method 1, repeatedly rotate copies by 180° around the midpoints of each shared edge, creating an interlocking plane tiling. In Method 2, build a grid using the Varignon parallelogram of DART and position copies onto alRead more
Draw non-convex quadrilateral DART with an internal dent angle exceeding 180°. In Method 1, repeatedly rotate copies by 180° around the midpoints of each shared edge, creating an interlocking plane tiling. In Method 2, build a grid using the Varignon parallelogram of DART and position copies onto alternating cells. Method 1 is generally preferred for its simplicity and directness without pre-drawing a grid.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-12/
See less