A triangle is cut by a line s parallel to the base in such a way that it divides the side of the triangle into parts in the ratio of 2 : 3. Find the other side of the triangle if it is known that the line s divides it into parts whose length is 5 cm.

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Answer to a math question A triangle is cut by a line s parallel to the base in such a way that it divides the side of the triangle into parts in the ratio of 2 : 3. Find the other side of the triangle if it is known that the line s divides it into parts whose length is 5 cm.

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Let's denote the sides of the triangle as follows: Let AB be the base of the triangle, and CD be the line parallel to the base AB that divides the side into parts in the ratio of 2:3. Let $ E $ be the point of intersection between CD and $ AB $. So, $ AE $ represents 2 parts and $ EB $ represents 3 parts. Given that the length of the line $ CD $ is 5 cm, we can express $ AE $ and $ EB $ in terms of this information. Let $ x $ be the length of $ AE $, then $ 5 - x $ will be the length of $ EB $. Since the parts are in the ratio 2:3, we can set up the following proportion: \[ \frac{x}{5-x} = \frac{2}{3} \] Now, cross-multiply to solve for $ x $: \[ 3x = 2(5 - x) \]

\[ 3x = 10 - 2x \]

\[ 5x = 10 \]

\[ x = 2 \] So, the length of $ AE $ is 2 cm, and the length of $ EB $ is $ 5 - 2 = 3 $ cm. Now, you know the lengths of $ AE $ and $ EB $. If $ AC $ is the other side of the triangle, then $ AC = AE + EC $. Substitute the values: \[ AC = 2 + 3 = 5 \] Therefore, the length of the other side of the triangle, AC, is 5 cm.

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