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How Can This Be True...?

Seems I stumped a few of you guys last week with a well placed RS pic. A tonne of emails followed asking me for the answer and as I had absolutely no idea it was lucky a few readers came to the rescue...

 

Darren wrote:
Subject: I'm a math nerd
In response to your 1st Random Shite image for this week:

The area of the colored shapes are:
Green triangle: 0.5 x 2 x 5 = 5 squares
Red triangle: 0.5 x 3 x 8 = 12 squares
Green hexagon: 8 squares
Yellow hexagon: 7 squares
The combined area: 5 + 12 + 8 + 7 = 32 squares

The 3 different sized triangles different slopes:
The green triangle has a slope of 2/5 (rise/run)
The red triangle has a slope of 3/8
The "multi-colored" triangles have a slope of 5/13

Because of the way the first shape is set up, the top of the "triangle" is concave (it bows inward).

If a straight line was drawn from the upper right corner to the lower left, there would be an "empty" gap of 0.5 squares. THUS, the first "triangle" has an area of 32 squares, although it appears to have an area of 32.5 squares (0.5 x 8 x 5).

Because of the way the second shape is set up, the top of the "triangle" is convex (it bows outward).

If a straight line was drawn from the upper right corner to the lower left, there would be a "filled in" gap of 0.5 square units. THUS, the second "triangle" has an area of 33 squares (with the "extra" square), although it appears to have an area of only 32.5 squares (0.5 x 8 x 5).

This is a visual representation of a common error in 4th grade arithmetic when students add fractions with unlike denominators: a/b + c/d = (a+c)/(b+d).

I'm sure you have enough math geeks who visit your site who will give you a
better description than I have. But hopefully this helps.

AcurA wrote:
Subject: funny triangles...
red and green triangles do not have the same angles (apart from the 90' one), they don't actually make a straight line if you look at it close enough.

<with held> wrote:
Subject: The two 'unequal' shapes on RS this week 8th March
The reason the two shapes are not the same is because the two triangles  are not 'similar triangle' and hence the apparent  straight line 'hypotenuse'  isn't a straight line but two lines with an angle in between them. In the first case the 'hypotenuse' starts off at a smaller angle and then changes to a larger angle at the 'join' whilst in the second case it is the bigger angle that starts off and the smaller on that takes over at the 'join'.

Hope that is not too technical for you to follow but if it is then just throw it away, I've had the pleasure of working it out logically which is a bonus on top of the laughs i get from the rest of the site. Keep up the good work, I'll keep rading it as long as I can strugle to my pc.


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