where $K$ is the area of the triangle. First, compute the semi-perimeter $s$:
Where Is $ K $ in the Triangle? Understanding Semi-Perimeter in Practical Math and Real-World Insights
Curious about how a single
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- Solution: A regular hexagon inscribed in a circle has its side length equal to the radius of the circle. Since the side length of the hexagon is 5 cm, the radius $r$ of the enclosing circular dome is also 5 cm. The circumference $C$ of a circle is given by:
- Question: A cartographer uses satellite data to map a triangular atoll with side lengths 13 km, 14 km, and 15 km. To optimize GPS signal reception, a circular transmission range must cover the entire triangle. Find the radius of the circumcircle of this triangle, express your answer in simplified radical form.
- Solution: We are given a triangle with sides $a = 13$, $b = 14$, and $c = 15$. The circumradius $R$ of a triangle with sides $a$, $b$, $c$ is given by the formula:
- Now use Herons formula to find the area $K$:
- Solution: Let the side length of the original equilateral triangle be $s$. The area $A$ of an equilateral triangle is:
- After reducing each side by 4 mm, the new side length is $s = 12$ mm. The new area is:
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$y = 0$: $25x^2 = 3600 \Rightarrow x^2 = 144$, $x = \pm 12$.
$y = \pm1$: $25x^2 = 144 + 3600 = 3744$, $x^2 = 3744/25 = 149.76$, not integer.
$y = \pm5$: $25x^2 = 144(25) + 3600 = 3600 + 3600 = 7200$, $x^2 = 288$, not square.
Eventually, only solutions are $(\pm12, 0)$.
Thus, exactly 2 lattice points.
Question: A high-performance computing simulation requires a grid size that is divisible by the least common multiple of all integers from 1 to 12. What is the smallest such grid size greater than 1 million?
Solution: We seek the smallest multiple of $\mathrm{lcm}(1,2,\dots,12)$ greater than 1,000,000.
Factor each number from 1 to 12:
$2^3$ (from 8),
$3^2$ (from 9),