To solve the problem of finding the length of the median from (A) to (BC) in triangle (ABC) where (AB=10), (AC=17), and (BC=21), we can use Apollonius's Theorem or coordinate geometry. Here's the step-by-step solution:

Step 1: Apply Apollonius's Theorem

Apollonius's Theorem states that for any triangle, the sum of the squares of two sides equals twice the square of the median to the third side plus twice the square of half the third side.

Let (AM) be the median from (A) to (BC) (where (M) is the midpoint of (BC)). Then:
[AB^2 + AC^2 = 2AM^2 + 2\left(\frac{BC}{2}\right)^2]

Step 2: Substitute the given values

• (AB=10), (AC=17), (BC=21)
• (\frac{BC}{2}=10.5)

[10^2 + 17^2 = 2AM^2 + 2(10.5)^2]
[100 + 289 = 2AM^2 + 2(110.25)]
[389 = 2AM^2 + 220.5]

Step 3: Solve for (AM)

[2AM^2 = 389 - 220.5 = 168.5]
[AM^2 = 84.25]
[AM = \sqrt{84.25} = \frac{\sqrt{337}}{2} \approx 9.18]

Answer: (\boxed{9.18}) (rounded to two decimal places) or (\boxed{\frac{\sqrt{337}}{2}}) (exact form).

Since the problem likely expects a numerical value, the rounded answer is (\boxed{9.18}).

(\boxed{9.18})

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