$\triangle R S T \sim \triangle R Y X$ by the SSS similarity theorem.
Which ratio is also equal to $\frac{R T}{R X}$ and $\frac{R S}{R Y}$ ?
$\frac{X Y}{T S}$
$\frac{S Y}{R Y}$
$\frac{R X}{X T}$
$\frac{S T}{r x}$
Answer (2)
The triangles △ RST and △ R Y X are similar by the SSS similarity theorem, leading to the ratio Y X ST being equal to RX RT and R Y RS . Therefore, the answer is Y X ST .
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The problem states that △ RST ∼ △ R Y X by the SSS similarity theorem.
By the definition of similar triangles, corresponding sides are proportional: R Y RS = Y X ST = RX RT .
The question asks for a ratio equal to RX RT and R Y RS .
The correct ratio is Y X ST .
Y X ST
Explanation
Understanding Triangle Similarity Since △ RST ∼ △ R Y X by the SSS 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 that we need to find a ratio that is also equal to RX RT and R Y RS . From the similarity relation, we know that Y X ST is also equal to RX RT and R Y RS .
Identifying the Correct Ratio Now, let's examine the given options:
TS X Y
R Y S Y
XT RX
Y X ST (Note: there was a typo in the original question. It should be Y X ST instead of r x ST )
Comparing these options with the ratio we found from the similarity, Y X ST , we can see that option 4, Y X ST , is the correct one.
Final Answer Therefore, the ratio that is also equal to RX RT and R Y RS is Y X ST .
Examples
Triangle similarity is a fundamental concept in geometry and has many practical applications. For example, architects use similar triangles to create scaled drawings of buildings. Surveyors use similar triangles to determine distances and heights of objects that are difficult to measure directly. In art, similar triangles can be used to create perspective and depth in paintings and drawings. Understanding triangle similarity helps in various fields that require scaling, measurement, and proportional reasoning.
