The numerical value of x in the expression for the side lengths of the triangle is 3.
The figure in the image depicts two similar triangles.
From the figure:
Side 1 of the smaller triangle = x + 3
Side 2 of the smaller triangle = 2
Side 1 of the larger triangle = x + 18
Side 2 of the larger triangle = x + 4
To solve for the value of x, we use proportion since the two triangles are similar:
(x + 3)/2 = (x + 18)/(x + 4)
Cross multiply:
2( x + 18 ) = ( x + 3 )( x + 4 )
2x + 36 = x( x + 4 ) + 3( x + 4 )
2x + 36 = x² + 4x + 3x + 12
2x + 36 = x² + 7x + 12
x² + 7x - 2x + 12 - 36 = 0
x² + 5x - 24 = 0
Next, factor using AC method:
( x - 3 )( x + 8 ) = 0
Equate each factor to 0 and solve for x:
( x - 3 ) = 0
x - 3 = 0
x = 2
( x + 8 ) = 0
x + 8 = 0
x = -8
Since we are dealing with dimensions, we take the positive value.
Hence, the value of x is 3.