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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Hester

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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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