# NCERT Mathematics Class 9 Exemplar Ch 8 Quadrilaterals Part 6

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**Exercise 8.4**

Q.1. A square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

Q.2. In a parallelogram ABCD, and . The bisector of ∠A meets DC in E. AE and BC produced meet at F. Find the length of CF.

Answer:

Q.3. P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which. Prove that PQRS is a rhombus.

Q.4. P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.

Q.5. P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which and Prove that PQRS is a square.

Q.6. A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.

Q.7. P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

Q.8. ABCD is a quadrilateral in which and Prove that and

Q.9. In Fig, , , and . Prove that and

Q.10. E is the mid-point of a median AD of ∆ABC and BE are produced to meet AC at F. Show that

Q.11. Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.

Q.12. E and F are respectively the mid-points of the non-parallel sides AD and BC of a

trapezium ABCD. Prove that and

Q.13. Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

Q.14. P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.

Q.15. ABCD is a rectangle in which diagonal BD bisects ∠ B. Show that ABCD is a square.

Q.16. D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.

Q.17. Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

Q.18. P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that and