Answer
Similar triangles states that if a triangles are similar then their corresponding ratios are in proportion.
As per the given statement:
[tex]\triangle ABC \sim \triangle ADE[/tex]
Since, these two triangles ABC and ADE are Similar then:
Corresponding sides are in proportion: i,e
[tex]\frac{AC}{AE} =\frac{BC}{DE}[/tex]
from the figure:
AC = x-1 units, AE = AC+CE = x-1+27 = x+26 units , BC = 20 units and DE = 3x+8 units
Substitute these values we get;
[tex]\frac{x-1}{x+26} =\frac{20}{3x+8}[/tex]
By cross multiply we get;
[tex](x-1)(3x+8) = 20(x+26)[/tex]
The distributive property says that:
[tex]a \cdot (b+c) = a\cdot b+ a\cdot c[/tex]
Using distributive property:
[tex]3x^2+8x-3x-8 = 20x +520[/tex]
Combine like terms;
[tex]3x^2+5x-8 = 20x +520[/tex]
or
[tex]3x^2+5x-8-20x-520=0[/tex]
Combine like terms;
[tex]3x^2-15x-528=0[/tex]
or
[tex]3(x^2-5x-176)=0[/tex]
[tex]x^2-5x-176=0[/tex]
Factorize the equation:
[tex]x^2-16x+11x-176=0[/tex]
[tex]x(x-16)+11(x-16)=0[/tex]
[tex](x-16)(x+11)=0[/tex]
By zero product property we have;
x-16 = 0 and x+ 11 = 0
x = 16 and x = -1
Since, side cannot be in negative
Therefore, the value of x = 16 units
