1. **State the problem:** We need to show that the median from the right angle vertex $C$ to the hypotenuse $AB$ in a right triangle is half the length of the hypotenuse. 2. **Recall the formula for the median to the hypotenuse:** The median from the right angle vertex to the hypotenuse equals half the hypotenuse length. 3. **Given points:** $A(5,5)$, $B(-3,-1)$, $C(1,-3)$. 4. **Calculate the length of the hypotenuse $AB$ using the distance formula:** $$AB = \sqrt{(5 - (-3))^2 + (5 - (-1))^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$$ 5. **Find the midpoint $M$ of $AB$:** $$M = \left(\frac{5 + (-3)}{2}, \frac{5 + (-1)}{2}\right) = (1, 2)$$ 6. **Calculate the length of the median $CM$:** $$CM = \sqrt{(1 - 1)^2 + (2 - (-3))^2} = \sqrt{0 + 5^2} = 5$$ 7. **Compare median length to half the hypotenuse:** $$\text{Half of } AB = \frac{10}{2} = 5$$ 8. **Conclusion:** The median from the right angle vertex $C$ to the hypotenuse $AB$ is $5$, which is exactly half the hypotenuse length $10$. This confirms the property that the median to the hypotenuse in a right triangle is half the hypotenuse.