Questions tagged [rectangles]

Questions about rectangles and their properties.

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Tiling a square with 3:1 rectangles

It is known that a square can be tiled with $n$ rectangles whose length is double their width for any $n > 4$. In particular, no two rectangles can overlap and no part of any rectangle is outside ...
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Rectangle on a Plane - how to find vertices

There is a Plane with equation ax + by + cz + d = 0 where a, b, c, d are known. There is a Rectangle on the above plane for which the length l and breadth b are known. Now two points P0 (x0, y0, z0) ...
57 views

Area of a rectangle inscribed in a semicircle as a function of $x$

The question given is, The figure shows a rectangle with two vertices on a semicircle of radius 2 and two vertices on the x-axis. Let $P(x, y)$ be the vertex that lies in the first quadrant. (a) ...
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Given $AB = 2$ and $AC = 3$, what are $AD$, $BD$, $m∠BAD$ and $m∠ABD$? [closed]

Given $AB = 2$ and $AC = 3$, what are $AD$, $BD$, $m∠BAD$ and $m∠ABD$?
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Calculate height and width of rotated rectangle

I hope this is an easy one. I need to calculate the new height and width of a scaled rotated rectangle. I know all its corners $(A, B, C, D)$, its rotation angle ($o$) and its center ($M$). Now one ...
126 views

Finding the rectangle of minimum area that can be divided into five rectangles such that the lengths of all their sides are different naturals.

Find the rectangle of minimum area that can be divided into five rectangles such that the lengths of all their sides are different natural numbers. I found a 11x11 rectangle such that when we divide ...
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The number of grid points p in a configuration of lines.

In some proof of theorem I have information Let A, B, and C denote the number of X, Y, and Z lines. By a classical results in geometry, the number of grid points p in a configuration with A, B and C, ...
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How do I calculate the new position of a given rectangle after subjecting it to rotation and scaling?

I want to calculate the new position of a modified version of a given rectangle. The given rectangle is shown in Yellow. The rectangle undergoing rotation and scaling is shown in Red. Step-1: The ...
44 views

Volume of 3D shape with rectangle as base [closed]

The base is rectangular with $4 \times 9.5$ dimensions. There are three edges extending at a $90^\circ$ angle from the base with lengths $0.2, 0.5$ and $0.3$. How would you calculate the total volume?...
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Find the size of inner rectangle rotated 45 degrees within another rectangle

As picture, when rotation is 45 degrees, there are multiple combinations of w and h, but if the ratio w/h is known, how do I calculate the size of inner rectangle? Thanks. P.S. the pink one is a ...
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How to find the vertices of a rectangle when the slope of the direction is given?

The title might be a bit confusing but I'll do my best to explain this. So this image is given: The coordinates of vertex A are (3,2). We also know that [AD] and [BC] measure 1 unit each and [AB] &...
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find a rectangle limited to 4 lines

I want to find center of rectangle given w = width and h = height. The rectangle is limited ...
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How to calculate the width and heights of three rectangles for a given frame

Three rectangles all of different sizes, two aligned left and one right Considering the image above, where: rectangles A, B and C must preserve their aspect ratio ...
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Partioning rectangles into rectangles and valid sub-rectangle extension rules.

Given a rectangle $A$ composed of unit squares, we then fuse grid squares into sub-rectangles $B_i$ in a way that the $B_i$ partition $A$. Example: XXO YZO YWW ...
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Rotated rectangle with dimensions $c \times d$ inscribed in a rectangle with dimensions $a \times b$. How to express $d$ knowing $a,b,c$?

What we know: length of $a$, $b$, $c$ $a$ and $b$ are the sides of the large rectangle what I need to find: length of $d$ or just one angle (yellow color) I found some solution on the internet with a ...
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How can I rotate a rectangle complete 360 degree

I implemented the logic to rotate rectangle $90$ degree but the logic is not working for complete $360$. It rotates until $90$ then goes in reverse direction until $180$ then again come back on $225$ ...
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Check if ellipse lies inside rectangle

I have Ellipse center Cx, Cy and radius (major radius Rx and minor radius Ry) with an angle of α (or α = rotation). Rectangle cordinates are (x1,y1), (x2,y2), (x3,y3) and (x4,y4). The Ellipse can be ...
33 views

Calculate rectangle inside ring parameters

Imagine we have a rectangle (with sides A, B) that fits into the ring (r1, R2). If r1, A and B are known, what is the equation for R2? (that is in my particular case) What are the interrelations ...
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Find the angle BEC in this quadrilateral - where have I gone wrong?

Question : I have received this question as a challenge from my teacher in Year 8 and I seem to make no progress. All I know is that $DC$ is parallel to $AB$, and that $DA$ is parallel to $AB$ ...
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Is there a simple way to know how many rectangles in some shaped grid there are if one knows the number of their corner points?

Is there a simple way to know how many rectangles in some shaped grid there are if one knows the number of their corner points? I.e. if I have some grid (possibly non-equilateral, other than ...
163 views

Most efficient way to pack circles with different radii in a rectangle of given size

Given a rectangle of size $x$, $y$, I would like to fit the maximum circles in it. The second rule is that my circles come in 3 different radii $r_1$, $r_2$, $r_3$, and I need the maximum number of ...
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Calculate remaining two corners of rectangle given two points and angle

I have a rectangle given two corners at opposite ends that we'll call A and B. I need to be able to figure out the remaining corners given these two points and the angle by which the rectangle is ...
57 views

Prove that a list of smaller rectangles form a cover of a rectangle as long as they satisfy two simple conditions

From the coding question Perfect Rectangles on leetcode. Given an array $\text{rectangles}$ where $\text{rectangles}[i] = [x_i, y_i, a_i, b_i]$ represents an axis-aligned rectangle. The bottom-left ...
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Area moments of parts of a rectangle divided by a line

I want to calculate the areas (and possibly their first and second moments) of a rectangle split into two parts by a line. It would be very helpful for me if there're closed form expressions for this ...
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Two dissection problems for rectangles

Let us consider two integer rectangles (that is, with sides of integer length) $S$ and $T$ of the same area. Then, obviously, $S$ can be dissected into several integer rectangles $A_1$, ..., $A_n$ (we ...
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Geometry : Ratio of area of a triangle to the rectangle containing it

$ABCD$ is a rectangle. $P,Q$ and $R$ are the midpoint of $BC$,$CD$ and $DA$. The point $S$ lies on the line $QR$ such that $SR:QS = 1:3$. The ratio of the triangle $APS$ and rectangle $ABCD$ is .... ? ...
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A rectangle P is divided into smaller rectangles by segments parallel to its sides. We call a point a t-point if its a vertex of two small rectangles. [closed]

A rectangle P is divided by segments parallel to its sides into smaller rectangles. We call a point a t-point if it is a vertex of exactly two such small rectangles. Prove that the number of t-points ...
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(Synthetic-)Geometrical theorems on the quadrature of a rectangle

I've been trying to find ways to prove the circle inversion theorem without the use of algebra, and the last part requires the equality between a square and a rectangle. I've managed to find two ...
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Numerically compute intersection of infinite line and rectangle

I'm trying to numerically compute where an infinite line defined by two points intersects a (finite) rectangle also defined by two points. Here's a illustration of the various ways that the line can ...
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Is there a specific name for parallel line segments that are the same length and aligned?

I'm trying to name an algorithm that identifies line segments with the following properties. They are parallel. They have the same length. Their endpoints are aligned along the perpendicular axis. ...