Questions about rectangles and its properties.

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Geometry Right triangles in a rectangle, find the area.

Please help, I've been struggling to figure out this problem for too long... Given the area of rectangle $ABCD = 1200 \text{ unit}^2$, find the area of right triangle $ABE$
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Points $S$ and $T$ are on side $\overline{CD}$ of rectangle $ABCD$ such $ \overline{AS}$ and $\overline{AT}$

Points $S$ and $T$ are on side $\overline{CD}$ of rectangle $ABCD$ such $\overline{AS}$ and $\overline{AT}$ trisects $\angle DAB$. If $CT = 3$ and $DS = 6$, then what is the area of $ABCD$? I have ...
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Find the area of shaded triangle inside of a rectangle.

In rectangle $ABCD$, $ P$ is the mid point of $AB$. $S$ and $T$ are the points of trisection of $DC$. If area of the rectangle is $70$ square units, with reference to the figure find area of shaded ...
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1answer
68 views

Working algorithm for testing two rectangles for overlapping in Earth GPS coordinates plain

Here is a seemingly simple, but actually quite tricky problem: I am trying to figure out the correct algorithm to test intersection/overlapping of two rectangles, which are plotted on the Earth's ...
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23 views

Filling the Unit Disk With Non-overlapping Rectangles

It intuitively seems to be true that no finite set of non-overlapping rectangles can fill the unit disk. Is this proposition really true? If so, how can one prove it?
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1answer
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How to determine coordinates on two different size rectangles

Hello community I hope all is well. I was wondering if someone could shine some light on the following problem. Let's say we have 2 rectangles (A and B) which are different in size. Let's say I have ...
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Making a Circle of Rotated Rectangles

By default, the system I'm using bases rectangle position by their center. If I copy a rectangle and rotate it, the result will look like this: Because I am trying to make a circle out of these ...
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1answer
61 views

How do I calculate the angle to place rectangles side by side around a circumference?

I have a circle with radius $r$ and several rectangular objects. All rectangles have the same sides $a$ and $b$ (although the side $a$ may be different from side $b$). I would like to place the ...
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1answer
29 views

Two-color square arrangement

There are $n$ green axis-parallel squares on the plane. You may scale and translate each square arbitrarily, as long as no two of them intersect. Now you have to put paiwise-disjoint red squares on ...
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51 views

Colorful squares arrangement

Can you arrange a finite number of green and red squares on the plane, sides parallel to the axes, such that: Every red square intersects $M$ green squares and no red square; Every green square ...
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1answer
93 views

9 rectangles have the same area as 20 squares

This is a fun little question that I encountered on a problem solving assessment: A small area is covered by 20 identical square tiles or 9 identical rectangular tiles. The length of the side of ...
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Least greedy square

There are $n$ squares of $m$ different colors. Squares of the same color are interior disjoint, but squares of different colors may intersect. For every square, define its "greed" as the maximum ...
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4answers
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Prove that between rectangles of a given $A$ area the square is the one with lower perimeter?

Prove that between rectangles of a given $A$ area the square is the one with lower perimeter? Im lost, cant even figure out what to do, or where to start?
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Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
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3answers
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Sum of the area of all the rectangles in a rectangular

We have a rectangular shape with the size n × m meters is divided into rectangles of size 1 × 1 meters. Question: Sum of the area of all the rectangles that can be seen in that rectangular is how ...
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149 views

Question on surface area and volume of cuboid

I came across a question: The surface area of the six faces of a rectangular solid are 4, 4, 8, 8, 18 and 18 square cms. The volume of the solid, in cubic centimetres is __. I can guess that 4 ...
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5answers
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Mensuration question about a hoop resting on a staircase

I came across a question recently: A hoop, as shown in the diagram, rests vertically at stair case. Note: AB = 12 cm, and BC = 8 cm. Find the radius of the hoop. Figure (hand-made): This is ...
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4answers
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Mensuration question

I recently came across a puzzling question: Two rectangles ABCD and DBEF are as shown in the figure. The area of DBEF is: Figure (hand-made): I know that through Pythagoras, we get ...
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Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
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2answers
73 views

Ellipse bounding rectangle

I'm trying to find the ellipse that bounds a rectangle in a way that the "distance" between the rectangle and the ellipse is the same vertically and horizontally. Here is an image to illustrate what ...
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2answers
355 views

how many rectangles in this shape

I've learned in my high school the solution to such riddle: How many rectangles are there in this shape: the solution is through combinations: in this shape is a $5\times 6$ grid so the number of ...
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2answers
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Disjoint rectangles with points at their corners

There is a set of $n$ points in the 2-dimensional plane. All x values and all y values are different. We want to draw the largest set of axis-parallel rectangles such that: All rectangles are ...
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Find Cathetus C1, C2 Knowing Hipotenuse or Find C1, C2, C3, C4 of Rectangle

I have a rectangle. I know all sides and 4 points for it (see black rectangle below). I resize one edge of this rectangle to any point (see resized red color rectangle and new point B). Here is the ...
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radius of circle inscribed in rectangle

I have two circles inside a rectangle(4 * 6), where the diameter of one of both is the total length of a side of the rectangle, and the other circle diameter is part of the length of the another side. ...
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2answers
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Derive formula for area of a circle from formula from area of rectangle

I need to explain how to derive the formula for the area of a circle from the formula for the area of a rectangle. The area of a rectangle is length(width) and the formula for the area of a circle is ...
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How can i figure out the points of a rectangle by just knowing the origin, width and length?

I've come across a mathematical problem, which I can't seem to solve with my limited geometry and trigonometry knowledge or by help of Wikipedia. I need to know the coordinate points of each corner ...
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Adjacency corner case

If two rectangles touch, but only touch at one of their corners (example, Rectangle A's upper-right corner is touch Rectangle B's lower-left corner), are they adjacent to one another? Why or why not?! ...
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Cutting rectangles while keeping the remainder connected

There is a cake with $n$ toppings. I want to cut a small piece out of each topping, such that the remaining cake is connected. Is this always possible? SOME FORMAL DETAILS (possibly not all of them ...
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Problem with finding “x” in triangle

I have got a problem with finding the x. I think the question isn't true or there should more informations on it.
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Tiling an L-shape with “almost square”s

ABSTRACT: Define an "almost square" as a rectangles with aspect ratio in $[{1 \over 2},2]$. What is the minimal number of interior-disjoint almost-squares required to tile the following L-shape (where ...
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Family of geometric shapes closed under division

The family of rectangles has the following nice properties: Every rectangle $R$ can be divided to two disjoint parts, $R_1 \cup R_2 = R$, such that both $R_1$ and $R_2$ are rectangles (i.e. belong ...
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2answers
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Largest bounded square

Suppose I have a triangular land-plot, but some part of it (the yellow part) is unusable. I want to build a square house on the usable (white) part. The house may be rotated (but must be square). What ...
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1answer
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cutting a cake without destroying the square toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned squares. The toppings may have different sizes, and they do not necessarily cover the entire cake. I want to divide the cake ...
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Does the rectangle contain the point?

A rectangle is defined by the 4 points ABCD. How can I tell if a given point, (x,y), is in the interior of the rectangle? My current guess is the following: To be inside the rectangle, the point ...
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1answer
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Approximate sector between two lines?

I need to approximate a red figure. I know coordinates of three points (little transparent circles). I also know a count of segments I need to divide this figure. The angle may be from 0 to Pi and ...
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KVPY Scholarship Exam Problem on finding the area of a rectangle

In a rectangle $ABCD$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $ABCD$ has equation $2x-y+4=0$. Then the area of the ...
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Linejoin for fat lines?

I draw a figure with 2 fat lines. I need to draw a join between these lines correctly. Long red lines are in a middle of each fat line. What I know: coordinates of white points. the angle between ...
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2answers
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What square does not contain the middle?

Consider the square $S = [-1,1]\times[-1,1]$. Suppose we put a smaller square inside it, which is rotated with an angle $\alpha$ relative to the large square. What is the largest such square that does ...
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212 views

cutting a cake without destroying the toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake. I want to ...
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1answer
69 views

Square coloring

There are 3 red axis-aligned interior-disjoint squares. There are 3 blue axis-aligned interior-disjoint squares. Is it always possible to find a pair of 1 red square and 1 blue square, such that ...
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114 views

rectangularizing the square

There is a square that I want to divide to n people, such that each person gets a rectangular piece with an equal area. An obvious option is to cut 1-by-n rectangles of size n-by-1, but the people ...
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What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. ...
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Maximal square covering

Let X be a shape in 2-dimensional space. Define a square covering of X as a set of axis-aligned squares, whose union exactly equals X. Note that some shapes don't have a finite square covering, for ...
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usable rectangles

This question deals with efficient division of land into land-plots. For the sake of this question, assume that a land-plot is usable only if it is a rectangle whose width/height ratio is between ...