A square is inscribed in a right triangle, as shown below. The legs of the triangle are 2 and 3. Find the side length of the square.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, E, F, G;
A = (0,0);
C = (5,0);
B = (3^2/5,3*4/5);
D = extension(B, A + (0,-5), A, C);
G = extension(B, C + (0,-5), A, C);
E = extension(D, D + (0,1), A, B);
F = extension(G, G + (0,1), B, C);
draw(A--B--C--cycle);
draw(D--E--F--G);
[/asy]
Can you help me with this?
P and q are complex numbers such that |p| = 2/5x + 2, |q| = −3/4x − 1 , and |p+q| = 2x + 4 .
On what interval must x fall?
a) [-2, -4/3]
b) (-∞, -4/3]
c) [-2, ∞)
d) [-4/3, -60/47]
I got the first one, but I don't know if it's right. If someone could help, that would be greatly appreciated!
