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10 Jan 2022, 11:21
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
37% (02:25) correct
63%
(02:53)
wrong
based on 27
sessions
History
Date
Time
Result
Not Attempted Yet
Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. It T is the mid point of CD, then the length of AT, in cm, is
A ) √15
B ) √13
C ) √12
D ) √14
E ) √16
A ) √15
B ) √13
C ) √12
D ) √14
E ) √16
10 Jan 2022, 11:59
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Because it is a regular hexagon, the vertically opposite sides (CD and FA will be 2 such sides) are parallel to each other.
1st) each interior angle of the regular hexagon is 120 degrees.
If we are to draw a diagonal from vertex C to vertex A, it will divide the interior angle into a 1 : 3 ratio.
Angle <ACT = 90 degrees
And
Angle <ACB = 30 degrees
(2nd) by drawing this perpendicular diagonal from vertex A to vertex C, we have created an isosceles triangle ABC with the sides: BC - AC - AB
The 2 equal sides are AB and BC, which are 2 of the side of the regular polygon. Each will have side length 2.
The angle between these 2 sides will be 120 degrees = interior angle of the regular hexagon
(3rd) if we draw a perpendicular height in triangle ABC between the 2 equal sides of the isosceles triangle, it will bisect the interior angle into 60 degrees and 60 degrees.
This height from the apex vertex B will meet AC at a 90 degree angle and Bisect AC (property of isosceles triangles)
Call this point X. X is the midpoint of AC
We then have created two 30-60-90 right triangles (one of which is triangle XCB) in which the hypotenuse is 2 (side BC)
The side across from the 60 degree angle will be XC. XC will have a length of sqrt(3)
AX will have the same length of sqrt(3) —-> giving the diagonal AC a length of:
AC = 2 * sqrt(3)
(4th) point T is on the midpoint of side CD. Thus, the length of TC will be (1/2) of 2
TC = 1
Triangle TCA is a 90 degree triangle within the regular hexagon with AT as its hypotenuse. At this point we can use the Pythagorean theorem to solve for length of AT.
(TC)^2 + (AC)^2 = (AT)^2
(1)^2 + [ 2 * sqrt(3) ]^2 = (AT)^2
1 + 4 * 3 = (AT)^2
AT = sqrt(13)
B
Posted from my mobile device
1st) each interior angle of the regular hexagon is 120 degrees.
If we are to draw a diagonal from vertex C to vertex A, it will divide the interior angle into a 1 : 3 ratio.
Angle <ACT = 90 degrees
And
Angle <ACB = 30 degrees
(2nd) by drawing this perpendicular diagonal from vertex A to vertex C, we have created an isosceles triangle ABC with the sides: BC - AC - AB
The 2 equal sides are AB and BC, which are 2 of the side of the regular polygon. Each will have side length 2.
The angle between these 2 sides will be 120 degrees = interior angle of the regular hexagon
(3rd) if we draw a perpendicular height in triangle ABC between the 2 equal sides of the isosceles triangle, it will bisect the interior angle into 60 degrees and 60 degrees.
This height from the apex vertex B will meet AC at a 90 degree angle and Bisect AC (property of isosceles triangles)
Call this point X. X is the midpoint of AC
We then have created two 30-60-90 right triangles (one of which is triangle XCB) in which the hypotenuse is 2 (side BC)
The side across from the 60 degree angle will be XC. XC will have a length of sqrt(3)
AX will have the same length of sqrt(3) —-> giving the diagonal AC a length of:
AC = 2 * sqrt(3)
(4th) point T is on the midpoint of side CD. Thus, the length of TC will be (1/2) of 2
TC = 1
Triangle TCA is a 90 degree triangle within the regular hexagon with AT as its hypotenuse. At this point we can use the Pythagorean theorem to solve for length of AT.
(TC)^2 + (AC)^2 = (AT)^2
(1)^2 + [ 2 * sqrt(3) ]^2 = (AT)^2
1 + 4 * 3 = (AT)^2
AT = sqrt(13)
B
Posted from my mobile device
28 Oct 2023, 10:08
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