1. **Problem statement:** Find the value of $1 - 20 = \text{?}$ with options involving angles $30^\circ$, $90^\circ$, $60^\circ$. Since the problem is unclear as written, we move to the next problem. 2. **Problem statement:** Given two complementary angles $x$ and $y$ with $\sin y = \frac{x}{5}$, find $\sin x$. - Complementary angles satisfy $x + y = 90^\circ$. - Using the identity $\sin y = \cos x$ because $y = 90^\circ - x$. So, $\sin y = \cos x = \frac{x}{5}$. We want $\sin x$. Since $\sin^2 x + \cos^2 x = 1$, then $$\sin x = \sqrt{1 - \cos^2 x} = \sqrt{1 - \left(\frac{x}{5}\right)^2}.$$ But $x$ is an angle, so $\frac{x}{5}$ is not a number unless $x$ is numeric. The problem likely means $\sin y = \frac{x}{5}$ where $x$ is numeric. Since the problem is ambiguous, we move to the next. 3. **Problem statement:** Given $\angle A = 70^\circ$ and $\angle 2x$ is an acute angle, find $x$. Since $2x$ is an acute angle, $0 < 2x < 90^\circ$. If $\angle A = 70^\circ = 2x$, then $$2x = 70^\circ \implies x = 35^\circ,$$ which is not among options. So likely $\angle A = 70^\circ$ and $\angle 2x$ is related differently. Since the problem is unclear, move to next. 4. **Problem statement:** Find the area of a square with vertices at $(1,1)$ and $(4,4)$. - The side length is the distance between these points along the square's side. Distance between $(1,1)$ and $(4,4)$: $$d = \sqrt{(4-1)^2 + (4-1)^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}.$$ If these points are opposite corners of the square, the side length $s$ satisfies: $$\text{diagonal} = s\sqrt{2} = 3\sqrt{2} \implies s = 3.$$ Area: $$s^2 = 3^2 = 9.$$ Answer: 9. 5. **Problem statement:** The line $2y = 3x + 6$ intersects the positive $y$-axis. Find the length of the intercept. - On the $y$-axis, $x=0$. Substitute $x=0$: $$2y = 6 \implies y = 3.$$ Length of intercept on positive $y$-axis is 3. 6. **Problem statement:** Find the radius of a circle with center at $(2,-3)$ passing through $(2,-1)$. - Radius is distance between center and point: $$r = \sqrt{(2-2)^2 + (-1 + 3)^2} = \sqrt{0 + 2^2} = 2.$$ Answer: 2. **Final answers:** 4) Area = 9 5) Intercept length = 3 6) Radius = 2