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# What are the lengths of sides NO and OP in triangle NOP belo

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What are the lengths of sides NO and OP in triangle NOP belo [#permalink]  26 May 2019, 01:44
Expert's post
What are the lengths of sides NO and OP in triangle NOP below?

Attachment:

#GREexcecise What are the lengths of sides NO and OP in triangle NOP below.jpg [ 9.76 KiB | Viewed 5676 times ]

[Reveal] Spoiler: OA
$$NO = 30$$ and $$OP = 10 \sqrt{34}$$

Math Review
Question: 8
Page: 260
Difficulty: medium

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Re: What are the lengths of sides NO and OP in triangle NOP belo [#permalink]  26 May 2019, 19:00
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Carcass wrote:
What are the lengths of sides NO and OP in triangle NOP below?

[Reveal] Spoiler: OA
$$NO = 30$$ and $$OP = 10 \sqrt{34}$$

Math Review
Question: 8
Page: 260
Difficulty: medium

Here,

Let us consider $$\triangle$$ NOP and $$\triangle MRP$$

the two $$\triangle$$are similar by A-A-A

Hence,

$$\frac{40}{50} = \frac{24}{NO}$$

NO = 30

Now in the $$\triangle NOP$$,

OP is the hypotenuse

therefore OP = $$\sqrt{(50^2 + 30^2)} = 10\sqrt{34}$$
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#GREexcecise What are the lengths of sides NO and OP in triangle NOP below.jpg [ 7.18 KiB | Viewed 5644 times ]

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Re: What are the lengths of sides NO and OP in triangle NOP belo [#permalink]  02 Jun 2019, 10:24
Expert's post
Carcass wrote:
What are the lengths of sides NO and OP in triangle NOP below?

Attachment:
#GREexcecise What are the lengths of sides NO and OP in triangle NOP below.jpg

[Reveal] Spoiler: OA
$$NO = 30$$ and $$OP = 10 \sqrt{34}$$

Let's add some more angles to the diagram AND also....

...add labels for points L and M

At this point, we can see that there are TWO SIMILAR triangles hiding in this diagram.
So, let's draw them apart to get a better idea of these two triangles:

What is the length of side NO?
KEEP CONCEPT: With any two SIMILAR triangles, the ratios of corresponding sides will be equal.

ON and LM are corresponding sides
Also, NP and MP are corresponding sides

Let x = the length of side NO

We can write: ON/LM = NP/MP
Replace values to get: x/24 = 50/40
Cross multiply: 40x = (24)(50)
Simplify: 40x = 1200
Solve: x = 1200/40 = 30

So, side NO has length 30
--------------------------------

What is the length of side OP?
Now that we know NO has length 30, we can see that we know two sides of triangle NOP
Since NOP is a RIGHT triangle, we can apply the Pythagorean Theorem.
So, we can write: 30² + 50² = OP²
Simplify: 900 + 2500 = OP²
Simplify: 3400 = OP²
Solve: OP = √3400 = = 10√34

So, side OP has length 10√34

Cheers,
Brent
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Re: What are the lengths of sides NO and OP in triangle NOP belo   [#permalink] 02 Jun 2019, 10:24
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# What are the lengths of sides NO and OP in triangle NOP belo

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