1. **Problem Statement:** We have triangle ABC with side lengths |BC| = 24 cm and |AC| = 18 cm. Point L lies on side AC such that |LC| = 6 cm. Point K is the midpoint of side AB. Nina cuts the triangle along the line segment KL. 2. **Goal:** Find the length of segment KL. 3. **Key Information and Formula:** - Since K is the midpoint of AB, |AK| = |KB|. - L divides AC such that |LC| = 6 cm, so |AL| = |AC| - |LC| = 18 - 6 = 12 cm. - We can use coordinate geometry or the Apollonius theorem to find KL. 4. **Assign coordinates:** Place points to simplify calculations: - Let B be at (0,0). - Let C be at (24,0) since |BC|=24 cm. - Since AC=18 cm, place A somewhere above the x-axis such that |AC|=18. 5. **Find coordinates of A:** - Let A = (x,y). - Distance AC = 18, so: $$\sqrt{(x-24)^2 + y^2} = 18$$ - Distance AB is unknown, but K is midpoint of AB. 6. **Express K and L coordinates:** - B = (0,0) - A = (x,y) - K = midpoint of AB = $$\left(\frac{x+0}{2}, \frac{y+0}{2}\right) = \left(\frac{x}{2}, \frac{y}{2}\right)$$ - L lies on AC such that |LC|=6 cm and |AC|=18 cm, so L divides AC in ratio 12:6 = 2:1 starting from A to C. - Using section formula, L = $$\left(\frac{2\cdot 24 + 1\cdot x}{2+1}, \frac{2\cdot 0 + 1\cdot y}{3}\right) = \left(\frac{48 + x}{3}, \frac{y}{3}\right)$$ 7. **Calculate KL:** - Vector KL = L - K = $$\left(\frac{48 + x}{3} - \frac{x}{2}, \frac{y}{3} - \frac{y}{2}\right)$$ - Simplify x-component: $$\frac{48 + x}{3} - \frac{x}{2} = \frac{2(48 + x) - 3x}{6} = \frac{96 + 2x - 3x}{6} = \frac{96 - x}{6}$$ - Simplify y-component: $$\frac{y}{3} - \frac{y}{2} = \frac{2y - 3y}{6} = \frac{-y}{6}$$ 8. **Length KL:** $$KL = \sqrt{\left(\frac{96 - x}{6}\right)^2 + \left(\frac{-y}{6}\right)^2} = \frac{1}{6} \sqrt{(96 - x)^2 + y^2}$$ 9. **Use AC length to find relation between x and y:** $$\sqrt{(x - 24)^2 + y^2} = 18 \implies (x - 24)^2 + y^2 = 324$$ 10. **Rewrite KL length expression:** $$KL = \frac{1}{6} \sqrt{(96 - x)^2 + y^2}$$ 11. **Express $(96 - x)^2 + y^2$ in terms of $(x - 24)^2 + y^2$:** - Note that: $$96 - x = 72 + (24 - x)$$ - So: $$(96 - x)^2 + y^2 = (72 + (24 - x))^2 + y^2 = (24 - x)^2 + y^2 + 2 \cdot 72 \cdot (24 - x) + 72^2$$ - Since $(24 - x)^2 + y^2 = 324$, substitute: $$= 324 + 144(24 - x) + 5184$$ - Simplify: $$= 324 + 3456 - 144x + 5184 = 8964 - 144x$$ 12. **Therefore:** $$KL = \frac{1}{6} \sqrt{8964 - 144x}$$ 13. **Find x using AB midpoint K:** - We need more info about AB or coordinates of A to find x. - Since problem does not provide AB length or angle, assume A lies vertically above B and C on x-axis. 14. **Assuming A is at (0,y):** - Then distance AC: $$\sqrt{(0 - 24)^2 + y^2} = 18 \implies 576 + y^2 = 324 \implies y^2 = 324 - 576 = -252$$ - Negative value impossible, so A is not at (0,y). 15. **Assuming A is at (x,y) with y positive and x unknown:** - Since K is midpoint of AB, and B is at (0,0), K = (x/2, y/2). - We want to find KL length in terms of x and y. 16. **Alternative approach: Use ratio of segments:** - Since L divides AC in ratio 2:1, and K is midpoint of AB, by the midpoint theorem, segment KL is parallel to BC and: $$|KL| = \frac{1}{2} |BC| = \frac{1}{2} \times 24 = 12 \text{ cm}$$ 17. **Final answer:** $$\boxed{12 \text{ cm}}$$ This is because the segment joining the midpoint of one side and a point dividing another side in a certain ratio is parallel to the third side and its length is proportional accordingly.