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# In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ

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Math Expert
Joined: 02 Sep 2009
Posts: 51185
In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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31 Jan 2016, 06:42
1
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Difficulty:

35% (medium)

Question Stats:

82% (01:46) correct 18% (02:19) wrong based on 78 sessions

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In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ = 17, what is the perimeter of MNP?

A. 102⁄3
B. 14
C. 16
D. 21
E. 26

Attachment:

2016-01-31_1840.png [ 5.86 KiB | Viewed 3564 times ]

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Joined: 02 Aug 2009
Posts: 7106
Re: In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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01 Feb 2016, 19:48
Bunuel wrote:

In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ = 17, what is the perimeter of MNP?

A. 102⁄3
B. 14
C. 16
D. 21
E. 26

Attachment:
2016-01-31_1840.png

Hi,
the geometry rule which one has to apply here is-
if midpoints of two sides of a triangle are joined, the line thus formed will be halfof and parallel to the third side..

now here all midpoints are joined, so each side will be half of the corresponding third side...
MEANS:- the perimeter of internal triangle is, thus, half of the perimeter of the outer triangle..

perimeter of outer triangle = 10+15+17=42..
D
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Re: In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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02 Feb 2016, 07:12
2
I used Mid point theorem to find the answer.

so
MN=0.5 YZ
NP= 0.5 XY
MP=0.5 XZ

so 5+8.5+7.5=21

D
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Re: In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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04 Feb 2016, 02:12
Bunuel wrote:

In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ = 17, what is the perimeter of MNP?

A. 102⁄3
B. 14
C. 16
D. 21
E. 26

Attachment:
2016-01-31_1840.png

At first, the question might look scary, but it can easily be solved by using the mid point theorem.

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

Using the theorem in this particular question, we know that the lengths of sides of the internal triangle will be half of the original
Therefore, MN = XZ/2
MP = YZ/2
NP = XY/2

From here, we can either solve for each side and then add or just divide the perimeter of the bigger triangle by 2
Therefore perimeter of MNP = 42/2 = 21 (Option D)
Manager
Joined: 03 Jan 2015
Posts: 79
In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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Updated on: 04 Feb 2016, 11:50
1
I guessed the answer which appears to be answer D. However, after researching the concepts that should be used to answer this question, I saw that I was right.

Anyway, explanation:
The triangle has 3 mid segments. A midsegment (e.g. MN) is always parallel to the third side of the triangle (XZ). The midsegment (MN) is always half the length of the third side (XZ).

Therefore, the perimeter of MNP is:
MN = 0.5XZ = 0.5 * 17 = 8.5
MP = 0.5YZ = 0.5 * 15 = 7.5
NP = 0.5XY = 0.5 * 10 = 5

8.5 + 7.5 + 5 = 21.0

Originally posted by saiesta on 04 Feb 2016, 05:20.
Last edited by saiesta on 04 Feb 2016, 11:50, edited 2 times in total.
Manager
Joined: 03 Jan 2015
Posts: 79
Re: In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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04 Feb 2016, 05:23
hwgmat2015 wrote:
I used Mid point theorem to find the answer.

so
MN=0.5 YZ
NP= 0.5 XY
MP=0.5 XZ

so 5+8.5+7.5=21

D

You got the right answer. However, you are corresponding the midpoints to the wrong third side.
MN = 0.5XZ not MN = 0.5YZ
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Re: In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ  [#permalink]

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03 Nov 2017, 10:13
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Re: In XYZ, points M, N, and P are midpoints. If XY = 10, YZ = 15, and XZ &nbs [#permalink] 03 Nov 2017, 10:13
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