Given a rectangle and three tangential circles embedded inside with known dimensions, find the length AB.
Answers
Answer:
7.34798 cm
Step-by-step explanation:
Consider A', B', and C' as the centers of the respective circles with diameter 3 cm, 6 cm, and 4 cm.
The length of segment A'C' is same as the sum of radii of the circles whose center these points represent, i.e. 1.5 cm + 2 cm, 3.5 cm.
As we see, points AC''C'A' form a parallelogram.
The length of AC'' is same as length of A'C' being the opposite sides of a parallelogram, i.e. 3.5 cm.
Now the distance between the two long sides of the rectangle is 6 cm as depicted by the diameter of the largest circle. The diameter of the medium circle is 4 cm, hence the shortest distance between the medium circle and the line segment AB is 2 cm. This gives the length of the segment CC' as 4 cm.
The length of segment CC''
= length of segment CC' - length of segment C'C''
= length of segment CC' - length of segment AA' (Opposite sides of parallelogram AC''C'A')
= 4 cm - 1.5 cm = 2.5 cm
For the right triangle AC''C,
Length of AC = Square root of (Square of length of AC'' - Square of length of CC'')
= Square root of ( (3.5*3.5) - (2.5*2.5) )
= Square root of ( 12.25 - 6.25 )
= Square root of ( 6 )
- Length of AC = 2.449 cm
On similar arguments, The length of segment B'C' is same as the sum of radii of the circles whose center these points represent, i.e. 3 cm + 2 cm, 5 cm.
As we see, points BC'''C'B' form a parallelogram.
The length of BC''' is same as length of B'C' being the opposite sides of a parallelogram, i.e. 5 cm.
As calculated previously, the length of the segment CC' as 4 cm.
The length of segment CC'''
= length of segment CC' - length of segment C'C'''
= length of segment CC' - length of segment BB' (Opposite sides of parallelogram BC'''C'B')
= 4 cm - 3 cm = 1 cm
For the right triangle BC'''C,
Length of BC = Square root of (Square of length of BC''' - Square of length of CC''')
= Square root of ( (5*5) - (1*1) )
= Square root of ( 25 - 1 )
= Square root of ( 24 )
- Length of BC = 4.89898 cm
Hence, The length of segment AB = Length of AC + Length of BC
= 2.449 cm + 4.89898 cm
The length of segment AB = 7.34798 cm
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Consider A', B', and C' as the centers of the respective circles with diameter 3 cm, 6 cm, and 4 cm.
The length of segment A'C' is same as the sum of radii of the circles whose center these points represent, i.e. 1.5 cm + 2 cm, 3.5 cm.
As we see, points AC''C'A' form a parallelogram.
A'c' = Ac" ( opposite sides of a parallelogram )
Now the distance between the two long sides of the rectangle is 6 cm as depicted by the diameter of the largest circle. The diameter of the medium circle is 4 cm, hence the shortest distance between the medium circle and the line segment AB is 2 cm. This gives the length of the segment CC' as 4 cm.
The length of segment CC''= length of segment CC' - length of segment C'C''
=> length of segment CC' - length of segment AA' (Opposite sides of parallelogram )
= 4 cm - 1.5 cm = 2.5 cm
For the right ∆AC''C,
Length of AC = ✓( AC''² - CC''² )
= ✓( (3.5*3.5) - (2.5*2.5) )
=> ✓( 12.25 - 6.25 )
=> ✓( 6 )
Length of AC = 2.449 cm
On similar arguments, The length of segment B'C' is same as the sum of radii of the circles whose center these points represent, i.e. 3 cm + 2 cm, 5 cm.
As we see, points BC'''C'B' form a parallelogram.
The length of BC''' is same as length of B'C' being the opposite sides of a parallelogram, i.e. 5 cm.
As calculated previously, the length of the segment CC' as 4 cm.
The length of segment CC''' = length of segment CC' - length of segment C'C'''
= length of segment CC' - length of segment BB' (Opposite sides of parallelogram BC'''C'B')
= 4 cm - 3 cm = 1 cm
For the right ∆ BC'''C,
Length of BC = ✓(BC'''² - CC'''²)
= ✓( (5*5) - (1*1) )
= ✓( 25 - 1 )
= ✓( 24 )
Length of BC = 4.89898 cm
Hence,
The length of segment AB = Length of AC + Length of BC
= 2.449 cm + 4.89898 cm
The length of segment AB = 7.34798 cm
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