Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
 If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?
1. Given a convex quadrilateral \(ABCD\) with perpendicular diagonals, we can use the theorem for orthodiagonal quadrilaterals. This theorem states that for such a quadrilateral, the sum of the squares of two opposite sides is equal to the sum of the squares of the other two opposite sides. Mathematically, this is expressed as:
   \[
   AB^2 + CD^2 = BC^2 + DA^2
   \]

2. We are given the side lengths:
   \[
   AB = 20, \quad BC = 70, \quad CD = 90
   \]

3. Substitute the given values into the equation:
   \[
   20^2 + 90^2 = 70^2 + DA^2
   \]

4. Calculate the squares of the given side lengths:
   \[
   20^2 = 400, \quad 90^2 = 8100, \quad 70^2 = 4900
   \]

5. Substitute these values back into the equation:
   \[
   400 + 8100 = 4900 + DA^2
   \]

6. Simplify the left-hand side:
   \[
   8500 = 4900 + DA^2
   \]

7. Isolate \(DA^2\) by subtracting 4900 from both sides:
   \[
   DA^2 = 8500 - 4900
   \]

8. Simplify the right-hand side:
   \[
   DA^2 = 3600
   \]

9. Take the square root of both sides to solve for \(DA\):
   \[
   DA = \sqrt{3600} = 60
   \]

Conclusion:
\[
\boxed{DA = 60}
\]