Dayami Rose

2022-07-01

Find all non-right angled dissimilar triangles having integer sides and integer area simuntaneously. Are there infinitely many such triangle?

SuefsSeeltHeRn8

Beginner2022-07-02Added 8 answers

$A=(0,0)$, $B=({n}^{2}-1,0)$, $C=(0,2n)$ with $n\ge 2$

Dayami Rose

Beginner2022-07-03Added 4 answers

Say you have a triangle with integer sides and area. Let the origin O be one vertex of your triangle, and place another vertex A at (a,0) (with a positive integer). It will be convenient if none of these are obtuse angles of the triangle, so for the argument's sake, assume OA is the longest side of the triangle.

Any third point B with coordinates (x,y) would yield a triangle OAB with integer area as long as y is an integer (base times height divided by 2) (if a is odd, y needs to be even as well). So we need to figure out what such points yield integer sides AB and OB. We have:

$|OB{|}^{2}={x}^{2}+{y}^{2}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|AB{|}^{2}=(a-x{)}^{2}+{y}^{2}={a}^{2}+2ax+{x}^{2}+{y}^{2}$

Any third point B with coordinates (x,y) would yield a triangle OAB with integer area as long as y is an integer (base times height divided by 2) (if a is odd, y needs to be even as well). So we need to figure out what such points yield integer sides AB and OB. We have:

$|OB{|}^{2}={x}^{2}+{y}^{2}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|AB{|}^{2}=(a-x{)}^{2}+{y}^{2}={a}^{2}+2ax+{x}^{2}+{y}^{2}$

Find an equation of the plane. The plane through the points (2, 1, 2), (3, −8, 6), and (−2, −3, 1), help please

A consumer in a grocery store pushes a cart with a force of 35 N directed at an angle of $25}^{\circ$ below the horizontal. The force is just enough to overcome various frictional forces, so the cart moves at a steady pace. Find the work done by the shopper as she moves down a $50.0-m$ length aisle.

??What is the derivative of $\mathrm{arcsin}\left[{x}^{\frac{1}{2}}\right]$?

What is the derivative of $y=\mathrm{arcsin}\left(\frac{3x}{4}\right)$?

Determine if the graph is symmetric about the $x$-axis, the $y$-axis, or the origin.$r=4\mathrm{cos}3\theta $.

How to differentiate $1+{\mathrm{cos}}^{2}\left(x\right)$?

What is the domain and range of $\left|\mathrm{cos}x\right|$?

How to find the value of $\mathrm{csc}74$?

How to evaluate $\mathrm{sec}\left(\pi \right)$?

Using suitable identity solve (0.99)raised to the power 2.

How to find the derivative of $y=\mathrm{tan}\left(3x\right)$?

Find the point (x,y) on the unit circle that corresponds to the real number t=pi/4

How to differentiate ${\mathrm{sin}}^{3}x$?

A,B,C are three angles of triangle. If A -B=15, B-C=30. Find A , B, C.

Find the value of $\mathrm{sin}{270}^{\circ}$.