Quadrilateral Diagonal Length Determination Using Triangle Inequality
The problem asks us to find the possible range for the length of the diagonal BD, denoted as \(x\), in a quadrilateral ABCD. We are given the lengths of the four sides: AB = 6 cm, BC = 18 cm, CD = 6 cm, and DA = 10 cm.
To solve this, we can consider the two triangles formed by drawing the diagonal BD. These triangles are \(\triangle ABD\) and \(\triangle BCD\). The diagonal BD acts as a common side for both these triangles.
We will apply the Triangle Inequality Theorem to each of these triangles.
Triangle ABD Analysis
The sides of \(\triangle ABD\) are AB, DA, and BD.
- Side AB = 6 cm
- Side DA = 10 cm
- Side BD = \(x\) cm
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Applying this theorem to \(\triangle ABD\):
- \(AB + DA > BD \implies 6 \text{ cm} + 10 \text{ cm} > x \implies 16 \text{ cm} > x\)
- \(AB + BD > DA \implies 6 \text{ cm} + x > 10 \text{ cm} \implies x > 10 \text{ cm} - 6 \text{ cm} \implies x > 4 \text{ cm}\)
- \(DA + BD > AB \implies 10 \text{ cm} + x > 6 \text{ cm} \implies x > 6 \text{ cm} - 10 \text{ cm} \implies x > -4 \text{ cm}\)
Since the length must be positive, the third inequality (\(x > -4\) cm) is always true. Combining the first two inequalities, we get the range for \(x\) based on \(\triangle ABD\) as:
\(4 \text{ cm} < x < 16 \text{ cm}\)
Triangle BCD Analysis
The sides of \(\triangle BCD\) are BC, CD, and BD.
- Side BC = 18 cm
- Side CD = 6 cm
- Side BD = \(x\) cm
Applying the Triangle Inequality Theorem to \(\triangle BCD\):
- \(BC + CD > BD \implies 18 \text{ cm} + 6 \text{ cm} > x \implies 24 \text{ cm} > x\)
- \(BC + BD > CD \implies 18 \text{ cm} + x > 6 \text{ cm} \implies x > 6 \text{ cm} - 18 \text{ cm} \implies x > -12 \text{ cm}\)
- \(CD + BD > BC \implies 6 \text{ cm} + x > 18 \text{ cm} \implies x > 18 \text{ cm} - 6 \text{ cm} \implies x > 12 \text{ cm}\)
Again, the second inequality (\(x > -12\) cm) is always true for a length. Combining the first and third inequalities, we get the range for \(x\) based on \(\triangle BCD\) as:
\(12 \text{ cm} < x < 24 \text{ cm}\)
Combining Triangle Inequalities for Diagonal BD
For the diagonal BD (\(x\)) to be a valid length in the quadrilateral ABCD, it must satisfy the conditions derived from both \(\triangle ABD\) and \(\triangle BCD\) simultaneously. We need to find the intersection of the two ranges:
- Range from \(\triangle ABD\): \(4 \text{ cm} < x < 16 \text{ cm}\)
- Range from \(\triangle BCD\): \(12 \text{ cm} < x < 24 \text{ cm}\)
To find the common range, we take the maximum of the lower bounds and the minimum of the upper bounds:
- Lower bound for \(x\): \(\max(4 \text{ cm}, 12 \text{ cm}) = 12 \text{ cm}\)
- Upper bound for \(x\): \(\min(16 \text{ cm}, 24 \text{ cm}) = 16 \text{ cm}\)
Therefore, the possible range for the length of the diagonal BD (\(x\)) is:
\(12 \text{ cm} < x < 16 \text{ cm}\)
This result indicates that the length of the diagonal BD must be strictly between 12 cm and 16 cm.
Similar Questions
What is the ratio of the area of the quadrilateral PQNM to the area of the quadrilateral RSNM?
- In a quadrilateral ABCD, AB = BC and CD = DA; AC and BD are diagonals such that AC = 6 cm and BD = 12 cm. What is the area of the quadrilateral?
Important Questions from Quadrilaterals
The ratio between the length and breadth of a rectangular park is 3 : 2. If a man cycling along the boundary at the speed of 12 km per hour completes one round in 8 minutes, then the area of the park in square meter will be
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ABCD is a trapezium in which AB || DC and DC is perpendicular to BC. If ∠DAB = 110°, then ∠ABC - ∠ADC =_____.
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