We need to find the length of the sides SR, RT, and the angle m∠TAS.
Finding the length of SR:
We consider that, in a rhombus, all of the sides are equal. So the sides ST and SR have to be equal:
[tex]ST=SR[/tex]
Substituting the values of these sides:
[tex]3x+30=8x-5[/tex]
We need to solve this equation for x.
Subtract 3x to both sides:
[tex]\begin{gathered} 30=8x-3x-5 \\ 30=5x-5 \end{gathered}[/tex]
Add 5 to both sides:
[tex]\begin{gathered} 30+5=5x \\ 35=5x \end{gathered}[/tex]
Divide both sides by 5:
[tex]\begin{gathered} \frac{35}{5}=x \\ 7=x \end{gathered}[/tex]
Now that we have the value of x, we can find the value of SR:
[tex]\begin{gathered} SR=8x-5 \\ \text{substituting x=7} \\ SR=8(7)-5 \\ \text{Solving the operations:} \\ SR=56-5 \\ SR=51 \end{gathered}[/tex]
SR=51
Finding the length of RT:
To find this length we need the value of z. And we can find the value of z considering that the diagonals bisect each other, so each side of a diagonal is equal to the other side of the diagonal. In this case, the blue and the red line in the image are equal:
We have that:
[tex]SE=AE[/tex]
Substituting the values of SE and AE:
[tex]3z=4z-8[/tex]
And now we solve for z by subtracting 3z to both sides:
[tex]\begin{gathered} 0=4x-3z-8 \\ 0=z-8 \\ \text{Add 8 to both sides:} \\ 8=z \end{gathered}[/tex]
With this value of z, we can find the length RT.
RT is the yellow line in the image:
Again we consider that the two sides of the diagonal are equal. Thus, RT is equal to:
[tex]\begin{gathered} RT=5z+5+5z+5 \\ \text{Combining like terms:} \\ RT=10z+10 \\ \text{substituting z=8} \\ RT=10(8)+10 \\ RT=80+10 \\ RT=90 \end{gathered}[/tex]
RT=90
Finally, we need to find the angle m∠TAS shown in blue in the image:
Since it is a rhombus, all of the angles in point E are equal to 90°:
So, considering only the triangle inside the rhombus marked in blue
We apply the property of triangles that tells us:
The sum of the internal angles of a triangle is equal to 180°.
So adding all of the blue angles we should get 180:
[tex]90+9y+8+5y-2=180[/tex]
And now we solve for y, first, combine like terms:
[tex]96+14y=180[/tex]
Subtract 96 to both sides:
[tex]\begin{gathered} 14y=180-96 \\ 14y=84 \end{gathered}[/tex]
Divide both sides by 14:
[tex]\begin{gathered} y=\frac{84}{14} \\ y=6 \end{gathered}[/tex]
And now that we have the value of y, we can find the value of m∠TAS:
[tex]\begin{gathered} m\angle TAS=9y+8 \\ \text{Substituting y=6} \\ m\angle\text{TAS}=9(6)+8 \\ m\angle TAS=54+8 \\ m\angle TAS=62 \end{gathered}[/tex]
m∠TAS=62°
Answer:
SR=51
RT=90
m∠TAS=62°
