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We are given a right-angled triangle \( \triangle ABC \), where \( \angle B = 90^\circ \). Points \( D \) and \( E \) are located on side \( BC \), trisecting it, with \( D \) near \( B \) and \( E \) near \( C \). We are asked to find the value of:

\[
\frac{8AE^2 - 3AC^2}{AD^2}
\]

### Step-by-Step Solution:

1. **Assign coordinates to the points:**
   - Since \( \triangle ABC \) is a right-angled triangle at \( B \), we can assume that \( B = (0, 0) \), \( A = (0, b) \), and \( C = (c, 0) \), where \( b = AB \) and \( c = BC \).

2. **Trisection of \( BC \):**
   - Since \( D \) and \( E \) trisect side \( BC \), the coordinates of these points can be computed.
   - The length of side \( BC = c \), and dividing it into three equal parts:
     - \( D = \left( \frac{2c}{3}, 0 \right) \)
     - \( E = \left( \frac{c}{3}, 0 \right) \)

3. **Find lengths \( AE \) and \( AD \):**
   - Using the distance formula, \( AE \) is given by:
     \[
     AE = \sqrt{ \left( 0 - \frac{c}{3} \right)^2 + \left( b - 0 \right)^2 } = \sqrt{ \frac{c^2}{9} + b^2 }
     \]
   - Similarly, \( AD \) is:
     \[
     AD = \sqrt{ \left( 0 - \frac{2c}{3} \right)^2 + \left( b - 0 \right)^2 } = \sqrt{ \frac{4c^2}{9} + b^2 }
     \]

4. **Substitute these into the given expression:**
   - Now, calculate \( AE^2 \) and \( AD^2 \):
     \[
     AE^2 = \frac{c^2}{9} + b^2, \quad AD^2 = \frac{4c^2}{9} + b^2
     \]
   - Substitute these into the expression \( \frac{8AE^2 - 3AC^2}{AD^2} \):
     \[
     \frac{8AE^2 - 3AC^2}{AD^2} = \frac{8 \left( \frac{c^2}{9} + b^2 \right) - 3c^2}{\frac{4c^2}{9} + b^2 }
     \]
     Simplifying the numerator:
     \[
     8AE^2 - 3AC^2 = 8 \left( \frac{c^2}{9} + b^2 \right) - 3c^2 = \frac{8c^2}{9} + 8b^2 - 3c^2 = \frac{-19c^2}{9} + 8b^ solution

In a right-angled triangle ABC, ∠B = 90°. D and E are points on side BC, trisecting it, where D is near to B and E is near to C. Find the value of the expression (8AE^2 - 3AC^2) / AD^2.

Learn how to solve a geometry problem involving a right-angled triangle, trisecting a side, and calculating the value of (8AE^2 - 3AC^2) / AD^2. This solution covers key geometry and algebra concepts, including the distance formula and trisection properties. Suitable for students in Grades 10-12.

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