Let T be an acute triangle. Inscribe a pair R, S of...

Let $T$ be an acute triangle. Inscribe a pair $R, S$ of rectangles in $T$ as shown : $=$ Let $A(x)$ denote the area of polygon $X$ find the maximum value (or show that no maximum exists), of $\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and $R, S$ over all rectangles as above.

Let $T$ be an acute triangle. Inscribe a pair $R, S$ of rectangles in $T$ as shown :
$=$
Let $A(x)$ denote the area of polygon $X$ find the maximum value (or show that no maximum exists), of $\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and $R, S$ over all rectangles as above.

Key Concepts

Properties of Rectangles

Rectangles, characterized by right angles and opposite sides that are equal and parallel, have unique properties that simplify area calculations and optimization approaches. Their properties are crucial for understanding how their dimensions can be adjusted within the confines of another shape, such as a triangle, to achieve an optimal layout.

Geometric Optimization

This concept involves determining the best possible configuration of geometric objects to maximize or minimize a given parameter—in this case, the area ratio. It requires setting up relationships between dimensions and using methods from calculus or geometric reasoning to find optimal values under given constraints.

Inscribed Figures

Inscribed figures refer to shapes that are drawn within another shape such that every vertex (or boundary) touches the enclosing shape. Here, the idea is critical because the rectangles are inscribed in the triangle, and their positioning significantly affects both their individual areas and the overall area ratio being optimized.

Area Ratios

The concept of area ratios involves comparing the sizes of different regions of a geometric figure. It is often used to study proportions, efficiency, or optimality in geometric configurations. In this context, the ratio of the summed areas of the two rectangles to the area of the triangle is analyzed to determine how well the triangles contain the inscribed rectangles.

Properties of Triangles

Understanding the fundamental properties of triangles, such as side lengths, angles, and area formulas, is essential when analyzing problems that involve inscribing figures within triangles. The type of triangle, particularly being acute in this problem, imposes additional geometric constraints which affect how other shapes can be optimally placed inside it.

Transcript

00:01 In this problem we need to show that all rectangle inscribed in a giving circle, square has the maximum area.

00:08 So for this purpose, i have just created a rough diagram and rectangle is a, b, c, d and its sides would be 2x and 2y.

00:21 Also, radius of the circle is, radius of the circle is nothing but a.

00:27 Now if we take this triangle that is a -o and this is suppose this is m.

00:39 So in triangle a -m -o we can use pythagodosuram.

00:44 So x -square plus y -square equals to a -square we can write easily.

00:51 Now if we talk about y, so y is nothing but a square minus x -square and its root.

00:59 Now we can find area and that is nothing but x into y and that is basically or area would be 4xy because side is 2x and 2y.

01:12 So area of rectangle is side into breath into length that is 2x into 2y.

01:18 That is 4xy.

01:20 So this would be 4x and y is nothing but root of a square minus x square.

01:28 Now for the further process we need to find a square because it will give some simplification to us because if a is maximum a square will be maximum.

01:42 A is minima so a square will be the same thing.

01:46 So this would be 16 x square times of a square minus x square basically.

01:55 So if we simplify this now we can consider this as some other function that is b.

02:00 Now we can find dv upon d x and that is that is simply we can multiply this also...

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