If $\triangle HLI \sim \triangle JLK$ by the SSS similarity theorem, then $\frac{ HL }{ JL }=\frac{ IL }{ KL }$ is also equal to which ratio?
A. $\frac{ HI }{ JK }$
B. $\frac{ HJ }{ JL }$
C. $\frac{ IK }{ KL }$
D. $\frac{ JK }{ HI }$
Answer (1)
Establish that corresponding sides of similar triangles have equal ratios.
Identify the corresponding vertices: H ↔ J , L ↔ L , and I ↔ K .
Express the equality of ratios: J L H L = L K L I = K J I H .
Conclude that the required ratio is: J K H I .
Explanation
Analyze the problem We are given that △ H L I ∼ △ J L K by the SSS similarity theorem. This means that the ratios of corresponding sides are equal. We are given that J L H L = K L I L . We need to find the other equal ratio.
Identify corresponding sides and ratios Since △ H L I ∼ △ J L K , the correspondence of vertices is H ↔ J , L ↔ L , and I ↔ K . Therefore, the corresponding sides are H L and J L , L I and L K , and I H and K J . Thus, we have the following equality of ratios: J L H L = L K L I = K J I H
Express the ratio using given options The ratio K J I H is the same as J K H I . Therefore, the ratio equal to J L H L = K L I L is J K H I .
State the final answer Thus, the ratio J L H L = K L I L is also equal to J K H I .
Examples
Understanding triangle similarity is crucial in architecture and engineering. For example, when designing a bridge, engineers use similar triangles to calculate heights and distances accurately. By knowing the ratios of corresponding sides in similar triangles, they can ensure the bridge is stable and meets the required specifications. This principle also applies to creating scaled models of buildings or other structures, where maintaining accurate proportions is essential for both aesthetics and structural integrity.
