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Difficulty:
45%
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Question Stats:
50% (02:10) correct
50%
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wrong
based on 6
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In parallelogram ABCD, the bisector of ABC intersects AD at P. If PD = 5 units, BP = 6 units, and CP = 6 units, what is the length of AB ?
(A) 10 units
(B) 8 units
(C) 7 units
(D) 4 units
(E) 2 units
(A) 10 units
(B) 8 units
(C) 7 units
(D) 4 units
(E) 2 units
19 Mar 2021, 01:37
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ABP is similar to PBC
\(\frac{6}{y} = \frac{y+5}{6}\)
\(y^2+5y-36=0\)
(y+9)(y-4)=0
y=4
Bunuel
19 Mar 2021, 07:07
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Correct answer D, using Angles and Similar Triangle Concepts. Awesome question working with Angles.
Draw the parallelogram. Starting from the lower, left vertex - label that Vertex A. From there label the Vertex immediately above it B and moving clockwise label the next 2 vertices C and D, respectively.
Draw Angle Bisector BP from Vertex B to a Point P on side AD
Angle <ABP = Angle <PBC ——> label these each of these equal angles as X degrees
Thus, the entire Angle bisected at vertex B = 2x degrees
Next draw a line connecting vertex C to point P.
This creates an Isosceles Triangle BPC within the parallelogram because we are told that BP = CP = 6. The opposite angles within this triangle will also be equal: Angle <PBC = angle <PCB = X
And since the Interior angles of any triangle sum to 180 degrees ——> angle <BPC = 180 - 2x
Next, using the theorem that states vertically opposite Angles in a parallelogram are equal —--> Angle at vertex D = 2x = Angle at vertex B
Using the theorem that Adjacent Angles in a parallelogram are Supplementary (Sum to 180 degrees), the Adjacent Angles in the parallelogram = 180 - 2x. These are the ——-> Vertically opposite angles at vertex A and Vertex C.
However, line CP divides vertex C into 2 angles——> <PCB and <PCD
<PCB + <PCD = 180 - 2x
Since <PCB = X ——-> angle <PCD = 180 - 3x
We have another Triangle created within the parallelogram: Triangle CPD
Since opposite, parallel sides of the parallelogram - sides BC and AD - are “cut” by the transversal PC———-> Alternate Interior Angles are equal ——-> angle <PCB = <CPD = X degrees
We also have a straight line Angle that sums to 180 degrees around point P: angle <CPD = X ——and—— angle <PCB = 180 - 2x
This means angle <APB = X degrees
We can now make two inferences:
(1)triangle ABP inside the parallelogram is an isosceles triangle with angle <ABP = angle <APB =X degrees. This also means the sides opposite the angles are equal ——> side AB = side AP. Call this side length Y
(2)triangle ABP and triangle BPC are similar isosceles triangles with corresponding interior angles that measure: X , X , and (180 - 2x)
Since we labeled AP = Y and we are given that PD = 5 ———> parallelogram side AD = AP + PD = (5 + y)
The opposite parallelogram side BC is equal with a length of 5 + y (rule: opposite sides of a parallelogram are equal)
We can now set up the corresponding sides of the similar triangles in an equal proportion-ratio:
In triangle APB:
-side AB = Y is opposite the angle of X degrees
-side BP = 6 is opposite the angle of (180
- 2x) degrees
In triangle BPC:
-side BP = 6 is opposite the angle of X degrees
-side BC = (5 + y) is opposite the angle of (180 - 2x) degrees
Y / 6 = 6 / (5 + Y)
Question is asking for the length of Side AB = Y = ?
Cross multiplying and setting the equation equal to 0, we end up with the quadratic equation of:
(Y)^2 + 5Y - 36 = 0
(Y + 9) (Y - 4) = 0
Positive length Y must = 4
Thus, the answer is Side AB = 4
(D)
Posted from my mobile device
Draw the parallelogram. Starting from the lower, left vertex - label that Vertex A. From there label the Vertex immediately above it B and moving clockwise label the next 2 vertices C and D, respectively.
Draw Angle Bisector BP from Vertex B to a Point P on side AD
Angle <ABP = Angle <PBC ——> label these each of these equal angles as X degrees
Thus, the entire Angle bisected at vertex B = 2x degrees
Next draw a line connecting vertex C to point P.
This creates an Isosceles Triangle BPC within the parallelogram because we are told that BP = CP = 6. The opposite angles within this triangle will also be equal: Angle <PBC = angle <PCB = X
And since the Interior angles of any triangle sum to 180 degrees ——> angle <BPC = 180 - 2x
Next, using the theorem that states vertically opposite Angles in a parallelogram are equal —--> Angle at vertex D = 2x = Angle at vertex B
Using the theorem that Adjacent Angles in a parallelogram are Supplementary (Sum to 180 degrees), the Adjacent Angles in the parallelogram = 180 - 2x. These are the ——-> Vertically opposite angles at vertex A and Vertex C.
However, line CP divides vertex C into 2 angles——> <PCB and <PCD
<PCB + <PCD = 180 - 2x
Since <PCB = X ——-> angle <PCD = 180 - 3x
We have another Triangle created within the parallelogram: Triangle CPD
Since opposite, parallel sides of the parallelogram - sides BC and AD - are “cut” by the transversal PC———-> Alternate Interior Angles are equal ——-> angle <PCB = <CPD = X degrees
We also have a straight line Angle that sums to 180 degrees around point P: angle <CPD = X ——and—— angle <PCB = 180 - 2x
This means angle <APB = X degrees
We can now make two inferences:
(1)triangle ABP inside the parallelogram is an isosceles triangle with angle <ABP = angle <APB =X degrees. This also means the sides opposite the angles are equal ——> side AB = side AP. Call this side length Y
(2)triangle ABP and triangle BPC are similar isosceles triangles with corresponding interior angles that measure: X , X , and (180 - 2x)
Since we labeled AP = Y and we are given that PD = 5 ———> parallelogram side AD = AP + PD = (5 + y)
The opposite parallelogram side BC is equal with a length of 5 + y (rule: opposite sides of a parallelogram are equal)
We can now set up the corresponding sides of the similar triangles in an equal proportion-ratio:
In triangle APB:
-side AB = Y is opposite the angle of X degrees
-side BP = 6 is opposite the angle of (180
- 2x) degrees
In triangle BPC:
-side BP = 6 is opposite the angle of X degrees
-side BC = (5 + y) is opposite the angle of (180 - 2x) degrees
Y / 6 = 6 / (5 + Y)
Question is asking for the length of Side AB = Y = ?
Cross multiplying and setting the equation equal to 0, we end up with the quadratic equation of:
(Y)^2 + 5Y - 36 = 0
(Y + 9) (Y - 4) = 0
Positive length Y must = 4
Thus, the answer is Side AB = 4
(D)
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