Math Solver

[Problem]
A right triangle has two legs both measuring 4 cm. What is the area of its circumscribed circle?

[Analysis]
This problem involves a right triangle and its circumscribed (circum) circle. For a right triangle the diameter of the circumcircle equals the length of its hypotenuse. Thus we will

1. compute the hypotenuse
2. find the radius as half the hypotenuse
3. compute the area of the circle using $\pi r^2$

[Solution]
Step 1 Compute the hypotenuse
$h = \sqrt{4^2 + 4^2}$
$h = \sqrt{16 + 16}$
$h = \sqrt{32}$
$h = 4\sqrt{2}$

Step 2 Find the radius of the circumcircle
$r = \frac{h}{2}$
$r = \frac{4\sqrt{2}}{2}$
$r = 2\sqrt{2}$

Step 3 Compute the area of the circle
$A = \pi r^2$
$A = \pi (2\sqrt{2})^2$
$A = \pi \cdot 4 \cdot 2$
$A = 8\pi$

[Answer]
The area of the circumscribed circle is $8\pi \text{ cm}^2$.

[Problem]
See attached files

[Image]
[User uploaded an image for reference]

[Analysis]
This is a system of two linear equations in two variables. We will use the elimination method to find the values of $x$ and $y$.

[Solution]
Step 1: Add the two equations to eliminate $y$
$(2x + y) + (2x - y) = 8 + 12$
$4x = 20$

Step 2: Solve for $x$
$4x = 20 \implies x = \frac{20}{4} = 5$

Step 3: Substitute $x = 5$ into the first equation to find $y$
$2 \times 5 + y = 8$
$10 + y = 8$

Step 4: Solve for $y$
$10 + y = 8 \implies y = 8 - 10 = -2$

[Answer]
$x = 5,\; y = -2$