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655-705 (Hard)|   Geometry|      
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Bunuel
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 06:54
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC such that AP = 6 cm. What is the length of BP?

A. 4
B. 6
C. 8
D. 10
E. 12

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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC Updated on: 12 Jun 2020, 07:17
Originally posted by RaghavKhanna on 12 Jun 2020, 07:02.
Last edited by RaghavKhanna on 12 Jun 2020, 07:17, edited 1 time in total.
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In triangle ABP:

BP < AP + AB
=> BP < 14

In triangle BPC

BP < PC + BC
=> BP < 6

only option A satisfies this.

=> Answer : A

Posted from my mobile device
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 07:08
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In Triangle ABC,
AB = AC = 8
BC = 4
AP = 6, PC = 6
∆ABP = AB+AP>BP
(Sum of two sides of triangle is greater than third side)
8+6>BP
BP < 14 -------(1)

∆CBP = CB+CP>BP
(Sum of two sides of triangle is greater than third side)
4+2>BP
BP < 6 -------(2)

Since BP is < 6&14 both. Only option that satisfy BP value = 4

IMO A

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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 07:11
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Bunuel
In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC such that AP = 6 cm. What is the length of BP?

A. 4
B. 6
C. 8
D. 10
E. 12
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PROPERTY: Sum fo any two sides in a Triangle > Third side

Please check the solution as attached here

Answer: Option A


Attachments

Screenshot 2020-06-12 at 7.40.17 PM.png
Screenshot 2020-06-12 at 7.40.17 PM.png [ 576.41 KiB | Viewed 8138 times ]

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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 07:31
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AC = AP + PC => PC = 8 - 6 = 2

In triangle BPC, BP < BC + PC => BP < 4 + 2 => BP < 6

Only option is A.
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 08:24
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The solutions above rely purely on estimates, using triangle inequalities, which is fine when the answer choices are this spread out, but would not work with different choices. The reason the answer is exactly 4 here is that the small triangle BPC, with sides 2, 4, x, is similar to the large triangle ABC with sides 4, 8, and 8. One way to see that: in general, if you have two lengths with a fixed angle between them, there's only one triangle you can possibly make. In this problem, we know, looking at triangle ABC, that if we put the angle C between a side of length 4 and a side of length 8, the opposite side is of length 8. But in triangle BPC, we have the same angle C between sides of length 2 and 4. Those are exactly half the lengths angle C sits between in triangle ABC, so triangle BPC is just half as big (in length) as ABC, and its third side is of length 4.

I'm using side-angle-side similarity (SAS) here, which is often used in trigonometry, but which I don't think I've ever needed for a real GMAT question.
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 08:57
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IanStewart
The solutions above rely purely on estimates, using triangle inequalities, which is fine when the answer choices are this spread out, but would not work with different choices. The reason the answer is exactly 4 here is that the small triangle BPC, with sides 2, 4, x, is similar to the large triangle ABC with sides 4, 8, and 8. One way to see that: in general, if you have two lengths with a fixed angle between them, there's only one triangle you can possibly make. In this problem, we know, looking at triangle ABC, that if we put the angle C between a side of length 4 and a side of length 8, the opposite side is of length 8. But in triangle BPC, we have the same angle C between sides of length 2 and 4. Those are exactly half the lengths angle C sits between in triangle ABC, so triangle BPC is just half as big (in length) as ABC, and its third side is of length 4.

I'm using side-angle-side similarity (SAS) here, which is often used in trigonometry, but which I don't think I've ever needed for a real GMAT question.
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I never knew you can use congruency to find out the ratio of sides!

So you are saying:
Traingle congruency: S A S
Big triangle: 4 angleC 8 --> third side 8
Small traingle: 2 angleC 4 --> therefore third side = 8/2=4
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 12 Jun 2020, 22:50
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AB = AC = 8
BC = 4
AP = 6 ; PC = 2
BP=?

In Tri. ABP,
1) BP < AB+ PA [ Property : Sum of two sides of the triangle is always greater than third side]
BP< 8+6
BP< 14
2) BP > 8-6 [ Property : Difference of two sides of the triangle is always less than the third side]
BP > 2

Thus, 2<BP<14 - equation 1

In Tri. BPC,
1) BP < BC+PC
BP< 4+2
BP < 6

2) BP> BC-PC
BP > 4-2
BP> 2

Thus , 2< BP< 6 - equation 2

From Eq1 & Eq2,
BP will be 3,4,5

Only option A suits the condition.

Answer : A
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 13 Jun 2020, 22:59
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Solved this by option matching. Clearly BP < BC+ PC. The Sum of 2 sides is greater than 3rd.
BP< 6
Only Option matching is A.
Attachments

1.PNG
1.PNG [ 19.01 KiB | Viewed 7498 times ]

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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 14 Jun 2020, 20:57
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Bunuel
In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC such that AP = 6 cm. What is the length of BP?

A. 4
B. 6
C. 8
D. 10
E. 12
Show more

In triangle ABP:

AP-AB < BP < AP + AB
=> 2 <BP < 14

In triangle BPC

BC- PC < BP < BC + PC
=> 2<BP < 6

only option A satisfies this.
A
is my Answer
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 15 Jun 2020, 07:16
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well,
i did the hard way. it is so funny to see the simple solutions.
i found the upper side of an isoceles trapezoide to b e 3.
bc is than being cut to 1/2:3:1/2
H of trapezoide is root 15/4 (WAS EASIER TO WORK WITH FRACTIONS)
so H^2+3.5^2=BP^2
SO BP^2 =64/4=16
BP=4. took me about 4-5 minutes.
But then i saw the other answers. i didnt even think of triangles sides.
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In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point оn side AC 04 Nov 2023, 06:29
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