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805+ (Hard)|   Geometry|      
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Bunuel
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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 16 Sep 2022, 04:23
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Note: Figure not drawn to scale.

In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*∠BCD = ∠BAD, what is the length of BC in cm?

(A) 12
(B) 13
(C) 19
(D) 20
(E) Cannot be determined

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vivek920368
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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 27 Sep 2022, 11:58
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Draw a parallel line to AB, name the point O, where it intersects BC.
OD = AB = 6.

since its a parallelogram AD=BO=7

now we need to find OC:

Angle BCD will be equal to Angle ODC, making OCD an isosceles triangle.
hence OC=OD=6

so BC = BO+OC = 7+6 = 13.

Ans = B
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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 20 Sep 2022, 11:37
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Bunuel

Note: Figure not drawn to scale.

In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*∠BCD = ∠BAD, what is the length of BC in cm?

(A) 12
(B) 13
(C) 19
(D) 20
(E) Cannot be determined

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If we make angle A 90 degrees, that makes angle C 45 degrees. Drawing a vertical line up from D, it intersects BC such that the left portion of BC is 7 and the right portion of BC is 6. Thus BC = 13.

If we make AB lean 30 degrees to the left of vertical, angle A is 120 degrees. That makes angle C 60 degrees. Drawing vertical lines up from A and from D, they intersect BC such that the middle portion of BC is 7 and the left and right portions of BC are each 3. Thus BC = 13.

If we make AB alllllmost horizontal, angle A is alllllmost 180 degrees. That makes angle C allllllmost 90 degrees. The total figure would be verrrrry flat and the length of BC would be basically the same as BA+AD = 13.

If we make angle A 60 degrees, that makes angle C 30 degrees. Drawing a vertical line down from B makes a 30-60-90 triangle. The hypotenuse is 6, so the shorter leg (the one along the bottom side of the figure) is 3 and the longer leg (the one that is the height of the figure) is \(3\sqrt{3}\). Drawing a vertical line straight up from D makes another 30-60-90 triangle. The shorter leg is the height of the figure and that's \(3\sqrt{3}\), so the longer leg is \(3\sqrt{3}\sqrt{3}\) = 9. The part of BC that is not included in the 9 is equal to the part of AD that is not included in the short leg of the first 30-60-90, so 7-3=4. Add that to 9 and we have 13.

Answer choice B.
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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 26 Sep 2022, 21:37
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Hi! can you please elaborate how did right part of BC become 6?

Thanks

ThatDudeKnows
Bunuel

Note: Figure not drawn to scale.

In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*∠BCD = ∠BAD, what is the length of BC in cm?

(A) 12
(B) 13
(C) 19
(D) 20
(E) Cannot be determined


If we make angle A 90 degrees, that makes angle C 45 degrees. Drawing a vertical line up from D, it intersects BC such that the left portion of BC is 7 and the right portion of BC is 6. Thus BC = 13.

If we make AB lean 30 degrees to the left of vertical, angle A is 120 degrees. That makes angle C 60 degrees. Drawing vertical lines up from A and from D, they intersect BC such that the middle portion of BC is 7 and the left and right portions of BC are each 3. Thus BC = 13.

If we make AB alllllmost horizontal, angle A is alllllmost 180 degrees. That makes angle C allllllmost 90 degrees. The total figure would be verrrrry flat and the length of BC would be basically the same as BA+AD = 13.

If we make angle A 60 degrees, that makes angle C 30 degrees. Drawing a vertical line down from B makes a 30-60-90 triangle. The hypotenuse is 6, so the shorter leg (the one along the bottom side of the figure) is 3 and the longer leg (the one that is the height of the figure) is \(3\sqrt{3}\). Drawing a vertical line straight up from D makes another 30-60-90 triangle. The shorter leg is the height of the figure and that's \(3\sqrt{3}\), so the longer leg is \(3\sqrt{3}\sqrt{3}\) = 9. The part of BC that is not included in the 9 is equal to the part of AD that is not included in the short leg of the first 30-60-90, so 7-3=4. Add that to 9 and we have 13.

Answer choice B.
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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 27 Sep 2022, 06:07
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Hi! can you please elaborate how did right part of BC become 6?

Thanks

ThatDudeKnows
Bunuel

Note: Figure not drawn to scale.

In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*∠BCD = ∠BAD, what is the length of BC in cm?

(A) 12
(B) 13
(C) 19
(D) 20
(E) Cannot be determined


If we make angle A 90 degrees, that makes angle C 45 degrees. Drawing a vertical line up from D, it intersects BC such that the left portion of BC is 7 and the right portion of BC is 6. Thus BC = 13.

If we make AB lean 30 degrees to the left of vertical, angle A is 120 degrees. That makes angle C 60 degrees. Drawing vertical lines up from A and from D, they intersect BC such that the middle portion of BC is 7 and the left and right portions of BC are each 3. Thus BC = 13.

If we make AB alllllmost horizontal, angle A is alllllmost 180 degrees. That makes angle C allllllmost 90 degrees. The total figure would be verrrrry flat and the length of BC would be basically the same as BA+AD = 13.

If we make angle A 60 degrees, that makes angle C 30 degrees. Drawing a vertical line down from B makes a 30-60-90 triangle. The hypotenuse is 6, so the shorter leg (the one along the bottom side of the figure) is 3 and the longer leg (the one that is the height of the figure) is \(3\sqrt{3}\). Drawing a vertical line straight up from D makes another 30-60-90 triangle. The shorter leg is the height of the figure and that's \(3\sqrt{3}\), so the longer leg is \(3\sqrt{3}\sqrt{3}\) = 9. The part of BC that is not included in the 9 is equal to the part of AD that is not included in the short leg of the first 30-60-90, so 7-3=4. Add that to 9 and we have 13.

Answer choice B.
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If you make AB 90 degrees, that makes the height of the figure 6. The triangle created by drawing a straight line up from D is a 45-45-90 (C is 45 degrees since it is half of A). In a 45-45-90, the legs are equal in length.

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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 24 Oct 2022, 21:59
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Bunuel

Note: Figure not drawn to scale.

In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*∠BCD = ∠BAD, what is the length of BC in cm?

(A) 12
(B) 13
(C) 19
(D) 20
(E) Cannot be determined

 


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Since the angles weren’t given, I took a chance and filled in favorable numbers that would give me a recognizable triangle to work with.

If we let <BCD = 60 ————> <BAD = 120

And since we have parallel lines, the interior angles along the same side of the transversal are supplementary (sum to 180
Degrees)

<ABC = 60

<ADC = 120

Then drop two perpendicular lines: one line from vertex A to a point X (AX) perpendicular to side BC —— creates a like segment BX on the unknown side

And a second line from vertex D to a point Y (DY) perpendicular to side BC —— creates a line segment YC on the unknown side

Both perpendicular lines create 30-60-90 right triangles.

(1st) right triangle AXB

Side BA is across from the 90 degree angle.

BX, across from the 30 degree angle, is (1/2) (6) = 3

And AX = 3 * sqrt(3)

(2nd) right triangle DYC

The other perpendicular line is parallel and equal to perpendicular line AX.

So DY = 3 * sqrt(3) ——— across from the 60 degree angle

YC, across from the 30 degree angle, has length = 3

(3rd) finally, the last piece of side BC is the parallel portion between the 2 perpendicular lines: XY

XY = 7, the length of the opposite side

In total, length of BC =

BX + XY + YC =
3 + 7 + 3 =

13

Note: not a foolproof option and there are better methods, but it’s a plan of action if stuck on such a hard question….

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In the figure above, AD||BC, AB = 6 cm and AD = 7 cm. If 2*BCD = BAD 13 Mar 2025, 11:04
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