A completely other, equally valid, way to come to the same answer. Note that CDIK is CEK - DEI therefor the area of CDIK = Area of CEK - Area of DEI For these areas we need the height of these triangles to line CE from both point K and I. Let's call the height of K to CE: line KL and the height of I: line IM Area CDIK = 1/2 * CE * KL - 1/2 * DE * IM. Since CE = 2 DE and DE = 5 this simplifies to 5/2 * (2KL - IM) to get KL and IM, I put the diagram in a coordinate system. I put point E in the origin and CE becomes the Y-axis and GE becomes the x-axis. line BE can now be written as y = -2x line GD can now be written as y = (1/2)x + 5 line GC can now be written as y = x + 10 point K is where BE and GC cross and therefore their y-coordinates are equal point I is where BE and GD cross and therefore their y-coordinates are equal The length of KL and IM is the absolute value of the difference between the x-coordinates of K and L and I and M respectively, but since the x coordinate of L and M is 0 this simplifies to the absolute value of the x coordinate of K and I respectively x-coordinate of I: -2x = (1/2)x + 5 5/2 x = -5 x = -2 IM = |-2| = 2 x-coordinate of K: -2x = x + 10 3x = -10 x = - (10/3) KL = |-10/3| = 10/3 Area CDIK = 5/2 * (2KL - IM) = 5/2 * (20/3 - 6/3) = 5/2 * 14/3 = 70/6 = 35/3 = 11 2/3 cm^2
CK is 1/3 of GC by construction. AC and GE are parallel lines at the end of GC. B and F are midpoints of these lines. By splitting line GC with lines AF and BE, you split line GC in thirds. Ancient Euclidian construction method. GC can be calculated using pythagoras to be 10√2 therefor KC is (10/3)√2. EDI is similar to GED. Both share angle D and a 90° angle. GD can be calculated using pythagoras, √(10^2 + 5^2) = √125 = 5√5, the ratio between the hypotenuses is 5√5 / 5 = √5. The ratio of the areas of both triangles is the square of the ratio of the sides. i.e. (√5)^2 = 5. GED is 1/2 * 10 * 5 = 25 --> Area DEI = 25/5 = 5 As my math teacher always said. "There is more than one road to Rome. Sometimes you want to take the scenic route."
Using coordinates and equations of straight lines you'll find C(10;10) D(10;5) I(8;4) K(20/3;20/3). Then use the shoelace formula and voilà : (700/3-630/3)/2=35/3.
A completely other, equally valid, way to come to the same answer.
Note that CDIK is CEK - DEI therefor the area of CDIK = Area of CEK - Area of DEI
For these areas we need the height of these triangles to line CE from both point K and I. Let's call the height of K to CE: line KL and the height of I: line IM
Area CDIK = 1/2 * CE * KL - 1/2 * DE * IM. Since CE = 2 DE and DE = 5 this simplifies to 5/2 * (2KL - IM)
to get KL and IM, I put the diagram in a coordinate system. I put point E in the origin and CE becomes the Y-axis and GE becomes the x-axis.
line BE can now be written as y = -2x
line GD can now be written as y = (1/2)x + 5
line GC can now be written as y = x + 10
point K is where BE and GC cross and therefore their y-coordinates are equal
point I is where BE and GD cross and therefore their y-coordinates are equal
The length of KL and IM is the absolute value of the difference between the x-coordinates of K and L and I and M respectively, but since the x coordinate of L and M is 0 this simplifies to the absolute value of the x coordinate of K and I respectively
x-coordinate of I:
-2x = (1/2)x + 5
5/2 x = -5
x = -2
IM = |-2| = 2
x-coordinate of K:
-2x = x + 10
3x = -10
x = - (10/3)
KL = |-10/3| = 10/3
Area CDIK = 5/2 * (2KL - IM) = 5/2 * (20/3 - 6/3) = 5/2 * 14/3 = 70/6 = 35/3 = 11 2/3 cm^2
CK is 1/3 of GC by construction. AC and GE are parallel lines at the end of GC. B and F are midpoints of these lines. By splitting line GC with lines AF and BE, you split line GC in thirds. Ancient Euclidian construction method. GC can be calculated using pythagoras to be 10√2 therefor KC is (10/3)√2.
EDI is similar to GED. Both share angle D and a 90° angle. GD can be calculated using pythagoras, √(10^2 + 5^2) = √125 = 5√5, the ratio between the hypotenuses is 5√5 / 5 = √5. The ratio of the areas of both triangles is the square of the ratio of the sides. i.e. (√5)^2 = 5. GED is 1/2 * 10 * 5 = 25 --> Area DEI = 25/5 = 5
As my math teacher always said. "There is more than one road to Rome. Sometimes you want to take the scenic route."
Using coordinates and equations of straight lines you'll find C(10;10) D(10;5) I(8;4) K(20/3;20/3).
Then use the shoelace formula and voilà : (700/3-630/3)/2=35/3.
35/3
only use similarity get 35/3
When you say find, do you mean calculate?
Yes 😃
Excellent. Because I found the blue area immediately but still have not yet calculated its area yet.
@@deningman The question on the next thumbnail is clearer because of you. Thank you. I hope you managed to find the blue area on this video!
@@deningman Old jokes are old.
35/3