Questions tagged [rectangles]
Questions about rectangles and their properties.
621 questions
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Find the length of the side DQ
In rectangle $ABCD$, points $P$ and $Q$ lie on sides $AD$ and $DC$ respectively, such that $AP = 2 \times DQ$.
Given that $AB = 5\,\text{cm}$, $BC = 10\,\text{cm}$, and the area of quadrilateral $BPQC$...
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What is the minimum number of line segments required to split a rectangle into the 12 free pentominoes?
Let $R_{m,n}$ be an $m$x$n$ rectangle where $m$ is the width and $n$ is the length, and $R_{m,n}$ is subdivided into a grid of unit squares.
A line segment is a horizontal or vertical segment drawn ...
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What is the smallest rectangle tileable by copies of each n-omino that can tile a rectangle?
If the wording of the question was a little confusing, here is the main idea:
Some $n$-ominoes cannot tile rectangles. We are excluding those.
Excluding those $n$-ominoes not tiling any rectangle, ...
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What's the minimum number of sets of n-ominoes needed for it to able to be tiled into a rectangle?
For any $n$, what's the minimum number of sets needed for all free $n$-ominoes to be able to be tiled in a rectangle?
For $n = 1$ and $n = 2$ the answer is 1 as there is only one monomino and domino.
...
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Find the length of the diagonal without trigonometry.
In this problem, where it is asked not to use trigonometry, we request the length of the diagonal of the large rectangle in which are inscribed two other identical rectangles separated by this ...
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How to show this simple equality without trigonometry?
Here is a simple question in which we are asked to express the diagonal of rectangle ABCD in terms of $|BH|$ and $|DH|$, where $[BH]$ is the height in the triangle $ABC$.
My answer is : $|BD|^2 = 3|BH|...
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Find the dimensions of the rectangle ABCD.
Here is a basic problem from my old book: $ABCD$ is a rectangle and $I\in BD$ belongs to the angle bisector of $\widehat{BAD}$. We have $AI=6$ and $CI=7$. The question is: what are the exact ...
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Rectangle and circle with same area and circumference
I saw a video proving that it is impossible to have a rectangle and a circle with the same circumference and area.
I noticed some symmetries and came up with this:
\begin{align*}
\text{1:} & \quad ...
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A subset of $\mathbb{R}^n$ that contains the projection on one of its point.
Let $ \mathbf{c} \in \mathbb{R}^n$. Define the function $\Pi^{\mathbf{c}}_k: \mathbb{R}^n \to \mathbb{R}^n$ such that $\Pi^{\mathbf{c}}_k(\mathbf{x})=(x_1,\ldots,x_k,c_{k+1},\ldots,c_n)$. The function ...
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Why is the area of the triangle $28 \text{ units}^2$?
I am trying to find the area of the shaded triangle inscribed in the rectangle, given that the area of this rectangle is $96$, YES it's $96$.
The diagram is shown below (albeit not accurately drawn),
...
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What is the maximum number of $a$-by-$b$ rectangles that will fit into a circle of radius $r$?
I have several round pieces of paper from which I need to cut as many rectangles as possible. The disc radius is 16 cm and the rectangles are 8 cm by 11 cm.
The ratio $16^2 \pi / (8 \times 11)$ of the ...
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Is there a standard way to define the union of aligned rectangles?
I am working with a collection of axis-aligned rectangles $R_1, R_2, \dots, R_h \subset \mathbb{Z}^2$, where each rectangle has integer coordinates. These rectangles are arranged in a grid-like ...
user969954
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The minimum rectangular sheet
What is the smallest possible area (in square decimeters) of a rectangular sheet of paper that can always be used to draw any two non-overlapping circles (they may be externally tangent to each other) ...
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Find all natural numbers n such that a rectangle of size 1995 by n can be divided into the tiles of 1 by k (pairwise different sized)
For any natural number $k$, we call a rectangle of size $1\times k$ a "long strip". Find all natural numbers $n$ such that a rectangle of size $1995\times n$ can be divided into a set of ...
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What is the ellipse that has largest area of its intersection with given rectangle if ellipse is allowed to cross one side of this rectangle?
We are given a rectangle and we want to find an ellipse that covers maximal possible area inside of rectangle if we allow one side of rectangle to be partly of completely inside ellipse while all ...
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