$\triangle R S T \sim \triangle R Y X$ by the SSS similarity theorem.
Which ratio is also equal to $\frac{R T}{R X}$ and $\frac{R S}{R Y}$ ?
$\frac{X Y}{T S}$
$\frac{S Y}{R Y}$
$\frac{R X}{X T}$
$\frac{S T}{Y x}$
Answer (2)
△ RST ∼ △ R Y X by the SSS similarity theorem, meaning corresponding sides are proportional.
Establish the proportional relationship: R Y RS = Y X ST = RX RT .
Analyze the given options to find the ratio equivalent to RX RT and R Y RS .
Conclude that the correct ratio is Y X ST .
Explanation
Understanding SSS Similarity Since △ RST ∼ △ R Y X by the SSS (Side-Side-Side) similarity theorem, it means that the corresponding sides of the two triangles are proportional. This gives us the following relationship: R Y RS = Y X ST = RX RT We are given the ratios RX RT and R Y RS , and we need to find which of the given options is also equal to these ratios.
Identifying Corresponding Sides From the similarity statement △ RST ∼ △ R Y X , we can identify the corresponding sides:
RS corresponds to RY
ST corresponds to YX
RT corresponds to RX
Therefore, the ratio of corresponding sides is: R Y RS = Y X ST = RX RT We are looking for a ratio that is equal to RX RT and R Y RS .
Analyzing the Options Now, let's examine the given options:
TS X Y : This is the inverse of X Y TS or Y X ST , so TS X Y = ST Y X . Since Y X ST = RX RT , then TS X Y is not equal to RX RT .
R Y S Y : This ratio involves sides that are not directly corresponding in the similarity statement.
XT RX : This ratio involves RX, which is a side of △ R Y X , and XT, which is not a side of △ R Y X or △ RST .
Y X ST : This ratio is exactly what we derived from the similarity statement: R Y RS = Y X ST = RX RT .
Final Answer Therefore, the ratio that is also equal to RX RT and R Y RS is Y X ST .
Conclusion The ratio that is also equal to RX RT and R Y RS is Y X ST .
Examples
Understanding similar triangles is crucial in architecture and engineering. For instance, 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 scaling architectural models, where maintaining proportional relationships is essential for accurate representation.
The ratio that is equal to both RX RT and R Y RS is Y X ST . This is directly derived from the similarity of the triangles △ RST ∼ △ R Y X .
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