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7 votes
Accepted

Area of tight-angled $\triangle POB$ given extensions of $OP,BP$ to circle centred at $O$ through $B$?

It is known that if two chords intersect, then product of their segments is the same (see Intersecting chords theorem). In the picture two chords intersect at point $P$, if you continue the vertical ...
Aig's user avatar
  • 5,511
3 votes

Find the value of $\int_{-\infty}^{\infty} \cos^{-1}\left(\cos\left(\frac{24+4 x^2}{4+x^2}\right)\right)dx$.

Note that $\frac{24+4x^{2}}{4+x^{2}} = \frac{8}{4+x^{2}}+4$ Now, when $x$ goes to infinity or negative of infinity, the expression tends to $4$, and when $x$ goes to $0$, expreession tends to 6. Thus,...
Debu's user avatar
  • 656
2 votes
Accepted

Area of the triangle inside the triangle

This problem is, as it comes to the reader, intriguing for more reasons. It is based on a figure, which is with intention a bad figure, so that the solver has no useful information from it, except for ...
dan_fulea's user avatar
  • 33.9k
2 votes

Area of the triangle inside the triangle

There is no solution to this problem. In order to show that, let's adjust a bit the figure: as shear mappings preserve areas, we can apply suitable shears to transform $MNP$ into an equilateral ...
Intelligenti pauca's user avatar
2 votes

Area of the triangle inside the triangle

Join $A$ to $N$, and let $$S_{\triangle AEN} = y \;\;\to\;\; S_{\triangle AMN} = 9 - y$$ Using that the ratio of base lengths of triangles with the same altitudes is the same as their ratio of areas, ...
John Omielan's user avatar
  • 48.8k
2 votes

Area of tight-angled $\triangle POB$ given extensions of $OP,BP$ to circle centred at $O$ through $B$?

If we could solve for the radius $r$, we would have enough information to compute $|\triangle OPB|$. We know $OP = r-5$ and $OB = r$. So $PB = \sqrt{r^2 + (r-5)^2}$ by the Pythagorean theorem. ...
heropup's user avatar
  • 139k
2 votes
Accepted

Find the area of a figure bounded by curves

The parametric equation of the ellipse is $ p(t) = (a \cos t , b \sin t) $ Plug this into the equation of the circle, then $ a^2 \cos^2 t + b^2 \sin^2 t = a b $ Using standard trigonometric identities,...
c'est pas normale's user avatar
1 vote

What is the reason for the ratios of square units not being the same as the ratios of units

When converting units, the conversion factor for area is the square of the conversion factor for length. Here’s how it works: Length: $1 \, \text{yard} = 3 \, \text{feet}$. Area: For a square with a ...
bruno's user avatar
  • 364
1 vote

A right-angled triangle has sides of integer length. Its area (in square metres) is twice its perimeter (in metres). What are the lengths of the sides

Let $a$ and $b$ be the legs. Then the area $ab/2$ doubles the perimeter $a+b+\sqrt{a^2+b^2}$. So $ab=4(a+b+\sqrt{a^2+b^2}).$ Multiply by the conjugate $a+b-\sqrt{a^2+b^2}$ and apply the difference of ...
Oscar Lanzi's user avatar
  • 40.1k
1 vote

Area of the triangle inside the triangle

$\newcommand{\f}{\frac} \newcommand{\c}{\cdot} \newcommand{\l}{\left(} \newcommand{\r}{\right)}$ I think, something is wrong with this problem. I’m getting that this picture doesn’t exist, because $x$ ...
Aig's user avatar
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