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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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
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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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