Length of CA = 6x - 30
Length of AT = 2x + 30
We know that all sides of a rhombus are congruent.
Which means :
[tex] = 6x - 30 = 2x + 30[/tex]
[tex] = 6x - 30 - 2x = 30[/tex]
[tex] = 4x - 30 = 30[/tex]
[tex] = 4x = 30 + 30[/tex]
[tex] = 4x = 60[/tex]
[tex] = x = \frac{60}{4} [/tex]
[tex] = x = 15[/tex]
Thus, the value of x = 15
Then :
Measure of CA :
[tex] = 6x - 30[/tex]
[tex] = 6 \times 15 - 30[/tex]
[tex] = 90 - 30[/tex]
[tex] = 60[/tex]
Thus, measure of CA = 60
Measure of AT :
[tex] = 2x + 30[/tex]
[tex] = 2 \times 15 + 30[/tex]
[tex] = 30 + 30[/tex]
[tex] = 60[/tex]
Thus, measure of AT = 60
Perimefer of rhombus CAT :
[tex] = 4 \times side[/tex]
[tex] = 4 \times 60[/tex]
[tex] = \color{plum}240[/tex]
Therefore, the perimeter of rhombus CAT = 240
