What pair of rectangle shapes is exchanged by cutting off a square?
What pair of rectangle shapes is exchanged by cutting off a square?According to https://oeis.org/A059966 the golden rectangle has period one when cutting off a square. What pair of rectangle (shapes)...
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You have $n$ rectangles of area $1$ (and variable height). Pack as many as...
You have $n$ rectangles of area $1$ (and variable height). Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$?How to...
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Finding safe points in polygons
Let $P$ be an axes-parallel polygon. A point $(x,y)\in P$ is called safe if for any pair $d_x\in[0,1],d_y\in[0,1]$, either $(x+d_x,y+d_y)$ or $(x-d_x, y-d_y)$ or both are in $P$.Figuratively, suppose...
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Find an angle in a rectangle. [closed]
A few days ago, I came across a problem that has driving me insane. It seems so simple, but I just can't solve it, it's like a blind spot to me. Here's the problem:A point $N$ is taken on the side $AB$...
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How can I check if a rotated rect intersects a non-rotated rectangle?
I have a rotated retangle and a rectangle. Both are defined by their corners (however, for the non-rotated ones I also have access to topLeft, topRight, etc).Is there a way I can check if they...
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Largest semicircle in a rectangle
We are given a rectangle. Its adjacent sides are $m$ and $n$ ($m \geq n$). We need to find the largest semicircle in this rectangle. How can we find it?EDIT: Beware of such situations!!
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Proving that no tile can fill both squares and equilateral triangles
Cut up a square into a finite number of identical tiles.Here is one possibility:How do I prove that the tiles could never be rearranged to form an equilateral triangle (with filled interior and no...
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Help me understand the answers
THE PROBLEMLet $ABCD$ be a rectangle and $E, F$ the means of the sides $AB$, respectively $AD.$ We know that there is a point $M$ on the segment $AC$ such that $\angle EMF = 60$ and $AC^2 = 4*ME*MF$ ....
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Geometry inequality question (picture attached) [closed]
This question is from a mock test of GRE Quant section. Unfortunately, they only provided the answer key without explanation.The question asks:Quantity $A = a + b$Vs.Quantity $B = 200 - c$Which is...
View ArticleGive a coordinate proof that a parallelogram is a rectangle iff its diagonals...
I'm kind of confused on how to solve this one, especially through using coordinate proof.I know how to prove the other way around. If you know that you have a rectangle, then you know that it's...
View ArticleLongest line in a rectangle [closed]
Given a rectangle, how can one show that the euclidian distance between any two points inside the rectangle (or on its borderies) is smaller or equal than the distance of the diagonal of that...
View ArticleFind all possible values of the perimeter of a rectangle with sides $x, y \in...
Find all possible values of the perimeter of a rectangle with sides positive integers $x$ and $y$ where its area is given by $A=3x+3y+\sqrt{9x^2+9y^2}$In other words if $xy=3x+3y+\sqrt{9x^2+9y^2}$,...
View ArticleMinimum Number of Rectangles Needed to Cover a set of square in an $\text{N}...
There is an $\text{N} \times \text{N}$ grid of squares. Some arbitrary subset of those squares must be covered by some number of rectangles (a given square must be covered entirely by one rectangle,...
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Area of one of four regions within a rectangle
There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region. Can you find the area...
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Is this a new point on the nine-point-circle of a triangle?
I was trying to get a feel for how to solve another question about the largest triangle that can fit in a unit square, by constructing the smallest enclosing square of a triangle in Geogebra. While...
View ArticleHow to calculate new position of a rectangle after translation and rotation?
I have a rectangle - lets say 100 long by 75 high. Origin been bottom left corner.I move the rectangle up and across by 10 and rotate by 3 degrees from centre of part.How do I calculate the new...
View Articlerectangular paddock, dimensions, maximise area it encloses
Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot,...
View ArticleValue of angles of a quadrilateral [closed]
$ABCD$ is a rectangle, $\overline{AC}$ and $\overline{BD}$ are its two diagonals, $O$ their intersection point, and $\angle COD=68^{\circ}$. What is the value of $\angle (BAO-OBC)$?
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Prove some results in a cube...
QuestionLet $ABCDA'B'C'D'$ be a cube and the points $M, N, Q$ the means of the sides $A'B', A'D', DC$. We denote by $\alpha=(MNQ)$.a) If the line $D'C'$ intersects the plane $\alpha$ at the point $T$,...
View ArticleMaximize rabbit pen size (no calc) [duplicate]
Question:To make an enclosure for your pet rabbits, you want to fence a rectangular pen against your house. Only 3 sides will need to be fenced. You have 170 feet of fencing material, and want to...
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