Answer:
[tex]x = \frac{2}{3}[/tex]
Step-by-step explanation:
Given
See attachment for triangles
Required
Find x
First, calculate tan A
From the first triangle:
[tex]tan\ A= \frac{x + 2}{x}[/tex]
Next, calculate tan B
From the second triangle:
[tex]tan\ B= \frac{4}{3x-1}[/tex]
[tex]tan\ A = tan\ B[/tex]
So, we have:
[tex]\frac{x + 2}{x}= \frac{4}{3x-1}[/tex]
Cross Multiply
[tex](x + 2)(3x - 1) = 4 * x[/tex]
Open brackets
[tex]3x^2 + 6x - x - 2 = 4x[/tex]
[tex]3x^2 + 5x - 2 = 4x[/tex]
Collect Like Terms
[tex]3x^2 + 5x - 4x- 2 =0[/tex]
[tex]3x^2 + x- 2 =0[/tex]
Expand
[tex]3x^2 + 3x - 2x- 2 =0[/tex]
Factorize:
[tex]3x(x + 1) - 2(x+ 1) =0[/tex]
Factor out x + 1
[tex](3x - 2)(x +1) = 0[/tex]
Split:
[tex]3x - 2 = 0\ or\ x + 1 = 0[/tex]
[tex]x = \frac{2}{3}\ or\ x = -1[/tex]
x can not be negative, so:
[tex]x = \frac{2}{3}[/tex]
