Find the Area of a Square Located Inside a Right Triangle - 4 Methods

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In this problem we explore 4 different ways of finding the area of the blue square inside a right triangle with the sides of 24 and 16 units.
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تمرين جميل . رسم واضح مرتب . شرح واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا . تحياتنا لكم من غزة فلسطين

اممدنحمظ
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Very clear exposition of the approaches. Please note that in English we read the decimal part as digits, ie.: 9.6^2 equals "ninety-two point one six", pi = "approximately three point one four", etc.

Ozymandi_as
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Excellent! I wud hv done just methods 1 and 2 :) Method 2 is the easiest...amazingly, the x-squared cancels out on both sides giving a simple eqn in x.

vjlaxmanan
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Method 5: If a and b are the legs of a right triangle (and c the hypotenuse), then x (a side of the square) can be calculated with this formula: x = (ab) / (a + b). In this case, x = (16 x 24) / (16 + 24) = 384 / 40 = 9.6. Just like found in Methods 1 to 4.

franzelman
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I liked all but the last was the most interesting because you introduced coordinate geometry thereby thinking outside the box😂

sharonmarshall
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Call the angle on the right of the triangle β. Tan β=16/24=3/2=x/(24-x) so 3x=2(24-x)=48-2x which simplifies to 5x=48 and so x=48/5. Area of square=x²=(48/5)²=2304/25=92, 16.

easy_s
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16 : 24 = 2 : 3 side of the blue square = 3a
2a+3a=16 5a=16 a=16/5

area of the blue square : 92.16

himo
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Similarity of triangles:

24/16 = ( 24-s ) / s
24/16 = 24/s -1
24/16+1 = 24/s
s = 24 / (24/16+1)
s = 9, 6 cm

Area = s²
Area = 92, 16 cm²

marioalb
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92.16 Answer
Let the square side = x,
then the yellow triangle on top length = 16-x, and the
the yellow triangle at the bottom base= is 24-x, and the area of the square = x^2; If you rotate the 16-x
side anti-clockwise, a new triangle is formed with side '16-x + 24-x's with a length of x; hence its area is
[16-x + 24- x][x]/2 = (40-2x)(x)/2 = 40x-2x^2/2 = 20x- x^2.
Since the area of the large rectangle is (16 x 24)/2 =192, then
20x-x^2 + x^2 =192 (since x^2 is the area of the square)
20x =192
x = 48/5 and
x^2 = (48/5)^2
= 2304/25
= 92.16 answer

devondevon
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Nice video shot, keep it up, thank you for sharing it :)

RixtronixLAB
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What if only the side of square is given and told to find hypotenuse (side of square is 1cm)

animeflez
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lets call the angle which is at the right side of the square by "a" and the angle at the top of the square by "b", they are complementary angles, so, the angles of the smaller triangle which is at the right side of the square are a, b and 90°, because a and b are complementary, and this also counts for the other small triangle, its angles are a, b and 90° too, so they are congruent, because we can identify the triangle congruence case ALA, am i wrong?

prod.shakr
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a variant on method 1
(16-x)/x = 16/24=2/3
16/x -1/=2/3
16/x=5/3
48=5x
96=10x
9.6=x
rather difficult to square without calculator. try this
(10-0, 4)²= 100-2(10×0.4) +0.4²
=100-8÷0.16
=92.16

davidseed
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We have a formula to find out x
Product of sides/sum of sides
24x16/(24+16)=48/5

crkr
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In competitive exams we have to make use of formulae, we dont have much time.

crkr
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Muy ilustrativos los métodos, pero entre el tercero y el cuarto, no tengo ganador son los mejores

icems.a.
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ar. of inscribed square92.16 sq unit.ans

adgfx
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(16-x)/x=16/24=2/3
48-3x=2x
5x=48
x=9.6

simonxinpan
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Nice!
∎=x(x) → ∆ → a = 16; b = 24 → x = ab/(a + b)

murdock
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L/(24-L)=(16-L)/L; LxL=(24-L)(16-L); 40L=24x16; L=24x16/40=9.6; LxL=92.16

santiagoarosam