- The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO = CODO . Show that ABCD is a trapezium.
- Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that: ar APB × ar CPD = ar APD × ar BPC
- Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that OA : OC = 3: 2. Is ABCD a parallelogram? Why or why not
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The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO = CODO . Show that ABCD is a trapezium.
Given: The diagonals of a quadrilateral ABCD intersect each other at the point O such that B O A O = D O C O i.e., C O A O = D O B O To Prove: A B C D is a trapezium Construction: Draw O E ∥ D C such that E lies on B C . Proof: In △ B D C, By Basic Proportionality Theorem, O D B O = E C B E . . . . . . . . . . . . ( 1 ) But, C O A O = D O B O (Given) . . . . . . . . . ( 2 ) ∴ From ( 1 ) and ( 2 ) C O A O = E C B E Hence, By Converse of Basic Proportionality Theorem, O E ∥ A B Now Since, A B ∥ O E ∥ D C ∴ A B ∥ D C Hence, A B C D is a trapezium.
Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that: ar APB × ar CPD = ar APD × ar BPC
IN quadrilateral ABCD, diagonal AC and BD intersect each other as P To prove : ar ( △ A P B ) × a r ( △ C P D ) = a r ( A P D ) × a r ( △ B P C ) Construction : Draw Al and CN perpendiculars on BD Proof: a r ( △ A P D ) × a r ( △ B P C ) = ( 1 2 × A L × D P ) × ( 1 2 × C N × B P ) = ( 1 2 × B P × A L ) × ( 1 2 × D P × C M ) = a r ( △ A P B ) × a r ( △ C P D )
Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that OA : OC = 3: 2. Is ABCD a parallelogram? Why or why not
Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that OA : OC = 3: 2. Is ABCD a parallelogram? Why or why not? Solution: In a It is given that AC and BD are the It intersects each other at the point O The properties of a parallelogram are a. The opposite sides of a parallelogram are b. The opposite sides of a parallelogram are equal. c. The opposite angles of a parallelogram are equal. d. The diagonals of a parallelogram bisect each other. e. Same-side interior angles supplement each other. f. The diagonals divide the parallelogram into two As OA: OC = 3: 2 is not equal ABCD is not a parallelogram Therefore, ABCD is not a parallelogram as OA is not equal to OC. ✦ Try This: Diagonals PR and QS of a quadrilateral PQRS intersect each other at O such that OP : OR = 5: 3. Is PQRS a parallelogram? Why or why not? ☛ Also Check: NCERT Exemplar Class 9 Maths Exercise 8.2 Sample Problem 4 Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that OA : OC = 3: 2. Is ABCD a parallelogram? Why or why not Summary: If diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that OA : OC = 3: 2, ABCD is not a parallelogram ☛ Related Questions: • • •
