In the given figure, \(DE \parallel BC\). If \(AD : AB = 1 : 3\) and \(AE = 2.5\text{ cm}\), then \(AC\) equals
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Remember that the ratio of the parts to the whole is constant across both sides of the triangle when a line is parallel to the base.
If the left side has a part-to-whole ratio of 1 to 3, the right side must also have a part-to-whole ratio of 1 to 3.
Thus, \(AC = 3 \times AE = 3 \times 2.5 = 7.5\text{ cm}\).
Step 1: State What Is Given. In triangle $ABC$, line $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. We are told $AD : AB = 1 : 3$ and $AE = 2.5$ cm. We need to find $AC$. Step 2: Apply the Basic Proportionality Theorem. Since $DE \parallel BC$, by the Basic Proportionality Theorem (Thales' Theorem): \[ \frac{AD}{AB} = \frac{AE}{AC} \] Step 3: Substitute the Known Ratio. We are given $\dfrac{AD}{AB} = \dfrac{1}{3}$. So: \[ \frac{1}{3} = \frac{AE}{AC} \] Step 4: Solve for AC. \[ AC = 3 \times AE = 3 \times 2.5 = 7.5 \text{ cm} \] Step 5: Understand the Reasoning. Since $D$ divides $AB$ in ratio $1:3$ from $A$, and $DE \parallel BC$, point $E$ also divides $AC$ in the same ratio $1:3$ from $A$. So $AE : AC = 1 : 3$, confirming $AC = 3 \times AE = 7.5$ cm. Step 6: Match with Options. 7.5 cm corresponds to option (1). \[ \boxed{7.5 \text{ cm}} \]
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