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The ratio of lengths of sides of a cyclic quadrilateral is 4 : 2 : 3 : 5. If one of the diagonals is twice the shortest side of the quadrilateral, what is the ratio of the length of the other diagonal to the longest side of the quadrilateral?
- a)10 : 9
- b)9 : 8
- c)11 : 10
- d)6 : 5
- e)4 : 3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The ratio of lengths of sides of a cyclic quadrilateral is 4 : 2 : 3 :...
In a cyclic quadrilateral, if the length of the four sides is a, b, c and d such that a and c form one pair of opposite sides while b and d form the other pair; while the length of the diagonals is p and q, then (p x q) = (a x c) + (b x d) Let the sides of the given cyclic quadrilateral measure 4x, 2x, 3x and 5jc units respectively.
As per the sequence, the sides measuring 4x and 3x are opposite sides while the sides measuring 2x and 5x are opposite sides.
Letp = 2 x smallest side = 2 x 2x = 4x units.
As per the sequence, the sides measuring 4x and 3x are opposite sides while the sides measuring 2x and 5x are opposite sides.
Letp = 2 x smallest side = 2 x 2x = 4x units.
(4x x q) = (4x x 3x) + (2x x 5x)
Hence option 3.
Most Upvoted Answer
The ratio of lengths of sides of a cyclic quadrilateral is 4 : 2 : 3 :...
Given:
- The ratio of lengths of sides of a cyclic quadrilateral is 4 : 2 : 3 : 5.
- One of the diagonals is twice the shortest side of the quadrilateral.
To find:
- The ratio of the length of the other diagonal to the longest side of the quadrilateral.
Solution:
Let's assume the lengths of the sides of the cyclic quadrilateral are 4x, 2x, 3x, and 5x respectively.
Properties of a Cyclic Quadrilateral:
- Opposite angles in a cyclic quadrilateral add up to 180 degrees.
- The opposite sides of a cyclic quadrilateral are supplementary.
Let's analyze the diagonals of the quadrilateral:
- Let AC and BD be the diagonals of the quadrilateral.
- Let O be the center of the circle circumscribing the quadrilateral, which is equidistant from all the vertices of the quadrilateral.
Using the properties of a cyclic quadrilateral:
- Since AC and BD are the diagonals of a cyclic quadrilateral, they intersect at O.
- Triangle AOB and triangle COD are similar triangles because they have the same angles (AOB and COD are opposite angles in the cyclic quadrilateral) and are in the same ratio (AO:CO = BO:DO = AO:DO = CO:BO).
Let's use the given information:
- One of the diagonals (let's assume AC) is twice the shortest side of the quadrilateral.
- Let the shortest side be 2x.
Using the properties of similar triangles:
- In triangle AOB, AO:OB = 2x:4x = 1:2
- In triangle COD, CO:OD = 2x:5x = 2:5
Let's find the ratio of the other diagonal to the longest side:
- The length of AC = 2(2x) = 4x
- The length of BD = 2x + 5x = 7x
- The length of the longest side = 5x
Therefore, the ratio of the length of the other diagonal to the longest side is 7x:5x, which simplifies to 7:5.
Answer:
The ratio of the length of the other diagonal to the longest side of the quadrilateral is 7:5, which is equivalent to 11:10. Therefore, the correct answer is option C) 11:10.
- The ratio of lengths of sides of a cyclic quadrilateral is 4 : 2 : 3 : 5.
- One of the diagonals is twice the shortest side of the quadrilateral.
To find:
- The ratio of the length of the other diagonal to the longest side of the quadrilateral.
Solution:
Let's assume the lengths of the sides of the cyclic quadrilateral are 4x, 2x, 3x, and 5x respectively.
Properties of a Cyclic Quadrilateral:
- Opposite angles in a cyclic quadrilateral add up to 180 degrees.
- The opposite sides of a cyclic quadrilateral are supplementary.
Let's analyze the diagonals of the quadrilateral:
- Let AC and BD be the diagonals of the quadrilateral.
- Let O be the center of the circle circumscribing the quadrilateral, which is equidistant from all the vertices of the quadrilateral.
Using the properties of a cyclic quadrilateral:
- Since AC and BD are the diagonals of a cyclic quadrilateral, they intersect at O.
- Triangle AOB and triangle COD are similar triangles because they have the same angles (AOB and COD are opposite angles in the cyclic quadrilateral) and are in the same ratio (AO:CO = BO:DO = AO:DO = CO:BO).
Let's use the given information:
- One of the diagonals (let's assume AC) is twice the shortest side of the quadrilateral.
- Let the shortest side be 2x.
Using the properties of similar triangles:
- In triangle AOB, AO:OB = 2x:4x = 1:2
- In triangle COD, CO:OD = 2x:5x = 2:5
Let's find the ratio of the other diagonal to the longest side:
- The length of AC = 2(2x) = 4x
- The length of BD = 2x + 5x = 7x
- The length of the longest side = 5x
Therefore, the ratio of the length of the other diagonal to the longest side is 7x:5x, which simplifies to 7:5.
Answer:
The ratio of the length of the other diagonal to the longest side of the quadrilateral is 7:5, which is equivalent to 11:10. Therefore, the correct answer is option C) 11:10.
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The ratio of lengths of sides of a cyclic quadrilateral is 4 : 2 : 3 : 5. If one of the diagonals is twice the shortest side of the quadrilateral, what is the ratio of the length of the other diagonal to the longest side of the quadrilateral?a)10 : 9b)9 : 8c)11 : 10d)6 : 5e)4 : 3Correct answer is option 'C'. Can you explain this answer?
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