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Ok so because ac is a diameter BD is half of chord DD'. Due to intersecting chords theorem, 5x = y² And because ABD is a right triangle: x² + y² = z² Substituting in the second given equation: 2(x² + 5x) = 468 x² + 5x = 234 x² + 5x - 234 = 0 x² + 18x - 13x - 234 = 0 x(x + 18) - 13(x + 18) = 0 (x - 13)(x + 18) = 0 x = 13 is the only positive result. x = 13 in. x + 5 = 2r: 13 + 5 = 2r r = 9 in. Area of the triangle = ½bh = ½xy = ½x√(5x) = 6.5√65 sq. in. Area of semicircle = ½πr² = ½π9² = ½81π = 40.5π sq. in. Area (Orange) = Area (Semicircle) - Area (Triangle) = 40.5π - 6.5√65 sq. in ≈ 74.83 sq. in.
X²+y²=z² 2z²=468 z²=234 z= √234 Draw a line from D to C, Triangles ADB is similar to ADC by proportion, x/√234= √234/(x+5) x²+5x=234 x²+5x-234=0 by factoring (x+18)(x-13)=0 x=-18 x=13 y²= z²-x² y²=234-169 y²=65 y= √65 Area ADB= ½xy = ½(13)(√65) =52.4 Area of semi circle ½π9² 81π/2 Area orange region 81π/2-52.4 = 74.83 sq in So, I think (x+y+z)²=1322 has no relevance in the problem
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Thanks for watching! Don't forget to like this video, subscribe to the channel, and leave a positive comment below! Also, if you want another test, please use the link to view the next video in this Math Skills Series:
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I agree, the first equation with 1322 isn't needed. Also, isn't the radius 23/2?
Ok so because ac is a diameter BD is half of chord DD'. Due to intersecting chords theorem,
5x = y²
And because ABD is a right triangle:
x² + y² = z²
Substituting in the second given equation:
2(x² + 5x) = 468
x² + 5x = 234
x² + 5x - 234 = 0
x² + 18x - 13x - 234 = 0
x(x + 18) - 13(x + 18) = 0
(x - 13)(x + 18) = 0
x = 13 is the only positive result.
x = 13 in.
x + 5 = 2r:
13 + 5 = 2r
r = 9 in.
Area of the triangle = ½bh
= ½xy = ½x√(5x)
= 6.5√65 sq. in.
Area of semicircle = ½πr²
= ½π9² = ½81π
= 40.5π sq. in.
Area (Orange) = Area (Semicircle) - Area (Triangle)
= 40.5π - 6.5√65 sq. in
≈ 74.83 sq. in.
X²+y²=z²
2z²=468
z²=234
z= √234
Draw a line from D to C,
Triangles ADB is similar to ADC
by proportion,
x/√234= √234/(x+5)
x²+5x=234
x²+5x-234=0
by factoring
(x+18)(x-13)=0
x=-18
x=13
y²= z²-x²
y²=234-169
y²=65
y= √65
Area ADB= ½xy
= ½(13)(√65)
=52.4
Area of semi circle
½π9²
81π/2
Area orange region
81π/2-52.4
= 74.83 sq in
So, I think (x+y+z)²=1322 has no relevance in the problem
yes, (x+y+z)²=1322 is irrelevant.