cutting a triangle to form a rectangle (Chile 1994 P2)

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0 replies

jlacosta

Apr 2, 2025

0 replies

Transformation of a cross product when multiplied by matrix A

Math-lover1 2

N 12 minutes ago by Math-lover1

I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$ ] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$ , but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?

2 replies

Math-lover1

Yesterday at 10:29 PM

Math-lover1

12 minutes ago

unfair coin, points winning 2024 TMC AIME Mock #9

parmenides51 5

N an hour ago by Math-lover1

Krithik has an unfair coin with a $\frac13$ chance of landing heads when flipped. Krithik is playing a game where he starts with $1$ point. Every turn, he flips the coin, and if it lands heads, he gains $1$ point, and if it lands tails, he loses $1$ point. However, after the turn, if he has a negative number of points, his point counter resets to $1$ . Krithik wins when he earns $8$ points. Find the expected number of turns until Krithik wins.

5 replies

parmenides51

Apr 26, 2025

Math-lover1

an hour ago

BrUMO 2025 Team Round Problem 15

lpieleanu 1

N 2 hours ago by vanstraelen

Let $\triangle{ABC}$ be an isosceles triangle with $AB=AC.$ Let $D$ be a point on the circumcircle of $\triangle{ABC}$ on minor arc $AB.$ Let $\overline{AD}$ intersect the extension of $\overline{BC}$ at $E.$ Let $F$ be the midpoint of segment $AC,$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AB}.$ Let the extension of $\overline{DG}$ intersect $\overline{AC}$ and the circumcircle of $\triangle{ABC}$ at $H$ and $I,$ respectively. Given that $DG=3, GH=5,$ and $HI=1,$ compute the length of $\overline{AE}.$

1 reply

lpieleanu

Apr 27, 2025

vanstraelen

2 hours ago

trigonometric functions

VivaanKam 9

N 2 hours ago by aok

Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!

9 replies

VivaanKam

Yesterday at 8:29 PM

aok

2 hours ago

No more topics!

cutting a triangle to form a rectangle (Chile 1994 P2)

parmenides51 3

N May 28, 2019 by brainiacmaniac31

Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.

3 replies

parmenides51

May 27, 2019

brainiacmaniac31

May 28, 2019

cutting a triangle to form a rectangle (Chile 1994 P2)

G H J

Reveal topic tags

geometry

rectangle

Triangle

G H BBookmark kLocked kLocked NReply

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parmenides51

30650 posts

parmenides51

#1 h May 27, 2019, 3:43 PM • 1 Y

Y by Adventure10

Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.

Z K Y

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Bookworm2604

208 posts

Bookworm2604

#2 h May 28, 2019, 7:49 AM • 2 Y

Y by Adventure10, Mango247

you can do it by the following procedure

Let ∆ABC be the triangle with angle(BAC) the biggest.
M and N be mid points of AB and AC respectively.
Foot of perpendicular from A on MN be D.
Cut along MN and AD.
Rotate ∆ADM in point M by 180° and ∆ADN in point N by 180° to get a rectangle.

Z K Y

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