I actually thought about the cosine theorem: (i) Use cosine theorem to deduce the angles of the yellow triangle. (ii) Having the angles of the yellow triangle, we can deduce the angles marked in the figure. (iii) Extend a line perpendicular to the top line intersecting the bottom common vertex of orange and yellow triangles to construct a right triangle. (iv) Since we can deduce the angles of the right triangle and we know the value of its hipotenuse, we can deduce all the other lengths, so we can obtaing the height of the orange triangle. (v) The line we have drawn earlier also defines a right triangle in the other side, so we can obtain all the lengths from the other triangle also. (vi) Finally, add up the bases of both right triangles (from the top line), multiply them by the height obtained (the line perpendicular to the top line), divide by 2 and that's it. Maybe a lil bit confusing, but it's hard with no letters at vertices...
Same but Andy's method is actually far more accurate. Since calculators use approximations for inverse trig functions, the solution I came to was actually only accurate to 2 decimal places while Andy's method resulted in the exact area.
It's been a long time since I've finished school but I don't remember learning Heron's formula. As for the TV Show I'm currently rewatching Dark. It's a German tv show which in my opinion is the most accurate SF about time travel and alternate time lines.
This is actually a very elegant way of doing it! I decided to go all out and do trig to do it, but it wasnt anywhere near as accurate or clean. I used the law of cosines to find the interior angles of the yellow triangle. Then I used a few identities (bottom left orange angle being equal to 180 - bottom right yellow angle, the top orange angle being equal to 180 - the top yellow angle all divided by 2, and the bottom right orange angle being equal to 180 minus the other 2 orange angles) to figure out the angles of the other triangle. Then I used the law of sines to get the length of the bottom side, then just computed ab*sin(C)/2 to get the area. The biggest problem with my method is that our computers use approximations for inverse trig functions, so it was only accurate to the 2nd decimal place.
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i used a different method. a) use cosine rule to determine angle between sides 8 and 4. (46.6) b) use again to determine angle between sides 6 and 4. (104.5) c) use this to determine left-most angles in orange triangle. (66.7 and 75.5 respectively) d) use sine rule to find length of the top side (6.33455532) e) use formula for area of triangle (0.5ab*sin(C)) to find the area, which comes to 11.618u².
Drop a perpendicular from the top vertex to the horizontal line that is common to both triangles. The perpendicular is a common height of both the yellow and orange triangles, and has length √(15). A right triangle is formed by this perpendicular, the side of length 8, and a segment of the horizontal line. Apply the Pythagorean theorem. The hypotenuse has length 8 and one side has length equal to the height, √(15). Doing the math, the third side has length 7, greater than the 6 for the horizontal side of the yellow triangle. So, the intersection between the perpendicular and the horizontal line is on the horizontal side of the orange triangle, not the yellow triangle. The yellow triangle is obtuse, not acute as shown in the figure. However, the solution shown is correct. Having an obtuse triangle may cause confusion in other solution methods.
The base of the orange triangle is also 6, as the distance towards the extended line drops in the ratio 8:4:0 Hence the area is the same, which as you said can be calculated with Heron.
It would be far better to try and draw to scale as it distracts some people who automatically recognise that the drawing is impossible.. The yellow triangle was drawn as an acute triangle when it is an obtuse triangle. Draw a line parallel to side "4". This then forms an isosceles triangle - same base angle and side lengths equal to 8. From similar triangles the parallel sides (8 and 4) mean the the triangle ratio is 2:1 hence the extended side "6" is equal to 12 which means X = 6. Use Heron's formulae to solve.
I used cosine theorem to find the angles that compose the top angle of the complete triangle. Than I marked the side of the orange triangle X and expressed the areas of the complete triangle and the orange triangle using X. Finally I subtracted the area of the yellow triangle from the complete triangle and compared it to the area of the orange triangle. After I found X I substituted it in the formula of the area using trig and found the area
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Currently taking a break from binging every season of the masked singer. This was hard to follow, but i have faith if i continue watching your videos i'll definitely pick up more of the stuff i don't currently understand.
In my quick look at the thumbnail I went a different route. First find the angle between the sides of length 4 and 8 and then take that from 180. Divide by 2 and you have the angle at the top of the orange triangle. Then find the angle between 4 and 6 and take away from 180 and you have the angle at the bottom left. Since you then have two angles and a side length I think the rest is pretty straightforward. Idk though I'm just sitting on the toilet
TV shows? I mostly watch RU-vid.😉 I struggled a little bit with this one and went down the full-on trig rabbit hole until I realized this was deceptively drawn. The angle between the l=6 side and the l=4 side is in fact >90 degrees. I rotated the diagram so that the diagonal line at the top was horizontal at the bottom, And drew vertical lines from the l=6, l=4 and the l=6, l=8 intersections down to the horizontal line. Since the two right angle triangles formed by those lines are proportional, it was a simple job to then work out their lengths and then compute the area; e.g., sqrt(135).
This is wrong on so many levels, if you take it as the problem is and it is to scale this is wrong. It is so wrong that you are just assuming that the two angles inside the golden triangle are equal, which they are not.
So isn't this whole solution wrong since a 4, 6, 8 triangle is an obtuse triangle whereas it is shown as an acute triangle? So the use of the angles and thales theorem are innacurate
Its the photo thats not accurate. I worked out all the interior angles and the angle opposite "8" is 104° approx, whereas the photo shows that under 90 degrees. They couldve maybe changed the triangles shape, as even in the final answer, the orange triangle is the same area as the yellow, but the orange looks so much bigger, so yeah its a misleading photo but he solved it right.
He may have solved it right for the info but the calculations said that the base of the second triangle is 6, but just based off of visuals it cannot be, because it is obvious, just like the two lines that were supposedly 4 each making 8 and the line did not bisect the edge in the middle
Honestly this doesn't look very interesting. It's pretty trivial to find all the angles using the cosine theoroem, than the sides with the sine theorem, and the height. But it will be pretty tiresome, Maybe you have a better way of doing it.
