To solve the problem, let's define what you're given and what we need to find.
You have:
- ( AB = x )
- ( BC = y )
- ( CA = z )
- ( A_1B_1 = 18 , \text{cm} )
- ( B_1C_1 = 21 , \text{cm} )
- ( A_1C_1 = 24 , \text{cm} )
- ( \frac{A_1B_1}{AB} = 2 )
From the ratio ( \frac{A_1B_1}{AB} = 2 ), we can express ( AB ) in terms of ( A_1B_1 ):
[
AB = \frac{A_1B_1}{2} = \frac{18 , \text{cm}}{2} = 9 , \text{cm}
]
Now, ( AB = x = 9 , \text{cm} ).
Next, we can use the lengths ( A_1B_1 ), ( B_1C_1 ), and ( A_1C_1 ) to find ( BC ) and ( CA ) using a similar relationship. First, let’s find an overall ratio for the smaller triangle ( A_1B_1C_1 ) to the larger triangle ( ABC ).
The sides of triangle ( A_1B_1C_1 ) are:
- ( A_1B_1 = 18 , \text{cm} )
- ( B_1C_1 = 21 , \text{cm} )
- ( A_1C_1 = 24 , \text{cm} )
The ratio of similarity between triangles ( A_1B_1C_1 ) and ( ABC ) can be denoted as ( k ). Since we have the length of ( AB ):
Using the known side ( A_1B_1 ):
[
k = \frac{A_1B_1}{AB} = 2
]
Now, we can find lengths ( BC ) and ( CA ) using ( k ):
For ( BC ):
[
BC = \frac{B_1C_1}{k} = \frac{21 , \text{cm}}{2} = 10.5 , \text{cm}
]
For ( CA ):
[
CA = \frac{A_1C_1}{k} = \frac{24 , \text{cm}}{2} = 12 , \text{cm}
]
So now we have all the lengths for triangle ( ABC ):
- ( AB = 9 , \text{cm} )
- ( BC = 10.5 , \text{cm} )
- ( CA = 12 , \text{cm} )
Thus, we conclude:
[
AB = 9 , \text{cm}, \quad BC = 10.5 , \text{cm}, \quad CA = 12 , \text{cm}
]
