1. **State the problem:** We need to find the length $VT$ in kite $RUST$ where diagonals $RT$ and $US$ intersect at $V$ at a right angle. 2. **Given information:** - $RU = 15$ - $RS = 41$ - $VS = 40$ - $UT = 49$ - Diagonals $RT$ and $US$ intersect at $V$ perpendicularly. 3. **Important properties:** - In a kite, the diagonals intersect at right angles. - The diagonal $US$ is horizontal, and $RT$ is vertical. - Point $V$ divides the diagonals into segments: $UV + VS = US$ and $RV + VT = RT$. 4. **Find $UV$:** Since $US = UV + VS$ and $VS = 40$, $UT = 49$ is given but $UT$ is not part of diagonal $US$, so we focus on $US$. 5. **Calculate $UV$:** $$UV = US - VS = (UV + VS) - VS = UV$$ But $US$ is not given directly; however, $UT = 49$ is a side, not diagonal. 6. **Use Pythagorean theorem on triangles formed by diagonals:** Since diagonals intersect at right angles, triangle $RUV$ and $TSV$ are right triangles. 7. **Calculate $UV$ using triangle $RUV$:** Given $RU = 15$ and $RV$ unknown, but $RS = 41$ is a side. 8. **Use the fact that $RS$ is a side of the kite:** Triangle $RVS$ is right angled at $V$ with legs $RV$ and $VS$ and hypotenuse $RS$. 9. **Calculate $RV$:** $$RS^2 = RV^2 + VS^2$$ $$41^2 = RV^2 + 40^2$$ $$1681 = RV^2 + 1600$$ $$RV^2 = 1681 - 1600 = 81$$ $$RV = 9$$ 10. **Calculate $VT$:** Since $UT = 49$ and $UV + VS = US$, and $UT$ is a side, we use triangle $TVU$ which is right angled at $V$. 11. **Calculate $UV$ using triangle $UVT$:** Triangle $UVT$ is right angled at $V$ with legs $UV$ and $VT$ and hypotenuse $UT$. 12. **Calculate $UV$:** Since $US = UV + VS$ and $US$ is diagonal, but $US$ is not given, we use $UT$ and $UV$. 13. **Calculate $UV$ using triangle $RUV$:** Triangle $RUV$ is right angled at $V$ with legs $RV=9$ and $UV$ unknown, hypotenuse $RU=15$. 14. **Calculate $UV$:** $$RU^2 = RV^2 + UV^2$$ $$15^2 = 9^2 + UV^2$$ $$225 = 81 + UV^2$$ $$UV^2 = 225 - 81 = 144$$ $$UV = 12$$ 15. **Calculate $VT$ using triangle $UVT$:** $$UT^2 = UV^2 + VT^2$$ $$49^2 = 12^2 + VT^2$$ $$2401 = 144 + VT^2$$ $$VT^2 = 2401 - 144 = 2257$$ $$VT = \sqrt{2257} \approx 47.52$$ 16. **Final answer:** Length $VT$ rounded to the nearest whole number is **48**.