1. **State the problem:** We need to find the length $KN$ in the figure with two right triangles sharing altitude $LN$ and given side lengths $KL=53$, $LM=75$, and $NM=60$. 2. **Identify the triangles and apply the Pythagorean Theorem:** The figure has two right triangles $KLN$ and $LNM$ with right angles at $N$. 3. **Use the Pythagorean Theorem formula:** For a right triangle with legs $a$, $b$ and hypotenuse $c$, the relation is $$a^2 + b^2 = c^2$$ 4. **Find $LN$ using triangle $LNM$:** Here, $LM$ is the hypotenuse, $LN$ and $NM$ are legs. $$LN^2 + NM^2 = LM^2$$ $$LN^2 + 60^2 = 75^2$$ $$LN^2 + 3600 = 5625$$ $$LN^2 = 5625 - 3600 = 2025$$ $$LN = \sqrt{2025} = 45$$ 5. **Find $KN$ using triangle $KLN$:** Here, $KL$ is the hypotenuse, $KN$ and $LN$ are legs. $$KN^2 + LN^2 = KL^2$$ $$KN^2 + 45^2 = 53^2$$ $$KN^2 + 2025 = 2809$$ $$KN^2 = 2809 - 2025 = 784$$ $$KN = \sqrt{784} = 28$$ 6. **Final answer:** The length $KN$ is 28 units.