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anemone
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A rectangle with sides $x$ and $y$ is circumscribed by another rectangle of area $A^2$. Find all possible values of $A$ in terms of $x$ and $y$.
Opalg said:[sp]Let $ABCD$ be the given rectangle and $PQRS$ a circumscribing rectangle, as in the diagram, where the angle $\theta = \angle PBA$ lies between $0$ and $\pi/2.$
Then $PQ = PB+BQ = x\cos\theta + y\sin\theta$ and $PS = x\sin\theta + y\cos\theta$. So the area of $PQRS$ is $$A^2 = ( x\cos\theta + y\sin\theta)(x\sin\theta + y\cos\theta) = xy(\sin^2\theta + \cos^2\theta) + (x^2+y^2)\sin\theta\cos\theta = xy + \tfrac12(x^2+y^2)\sin2\theta.$$ Thus the minimum value of A^2 is $xy$ (when $\theta=0$ or $\pi/2$) and the maximum value is $xy + \frac12(x^2+y^2) = \frac12(x+y)^2$ (when $\theta = \pi/4$). Finally, $A$ takes all values from $\sqrt{xy}$ to $\frac1{\sqrt2}(x+y)$ inclusive.[/sp]
Geometry is the branch of mathematics that deals with the study of shapes, sizes, positions, and dimensions of objects. Trigonometry is the branch of mathematics that deals with the study of triangles and the relationships between their sides and angles.
Geometry and trigonometry have many practical applications in everyday life, such as measuring distances, angles, and areas of objects and structures. They are also used in fields such as architecture, engineering, and navigation.
Some common formulas used in geometry include the Pythagorean theorem, the area and perimeter formulas for different shapes, and the volume and surface area formulas for 3D objects. In trigonometry, common formulas include the sine, cosine, and tangent ratios, the law of sines and cosines, and the double and half angle formulas.
Practice is key to improving skills in geometry and trigonometry. Consistently solving problems and understanding the concepts will help build a strong foundation. Seeking out additional resources such as textbooks, online tutorials, and practice tests can also aid in improving skills.
Yes, geometry and trigonometry are not only used in mathematics but also have applications in other areas of science such as physics, astronomy, and geography. They are used to model and understand various natural phenomena and are essential tools for scientific research and analysis.