Discover new perspectives and gain insights with IDNLearn.com's diverse answers. Get accurate and timely answers to your queries from our extensive network of experienced professionals.
Sagot :
To solve the problem, let's start by understanding the concept of similar triangles and the SSS (Side-Side-Side) similarity theorem. According to the SSS similarity theorem, if two triangles are similar, the ratios of the lengths of their corresponding sides are equal.
Given two triangles, [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], we are told they are similar. This means:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} \][/tex]
We need to determine which of the given ratios are equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Let's go through each of the options:
1. [tex]\(\frac{XY}{TS}\)[/tex]
To be a valid ratio for similar triangles, the sides in this ratio must be corresponding sides of the similar triangles. The sides [tex]\(XY\)[/tex] and [tex]\(TS\)[/tex] are not corresponding sides between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex]. Therefore, this ratio is not equivalent to [tex]\(\frac{RT}{RX}\)[/tex].
2. [tex]\(\frac{SY}{RY}\)[/tex]
Here, the side [tex]\(SY\)[/tex] is involved, but it is not mentioned which triangle's corresponding side ratio it is equivalent to. Additionally, [tex]\(RY\)[/tex] is one of the original sides given, and it does not establish the necessary proportionality with [tex]\(RT\)[/tex] or [tex]\(RX\)[/tex]. Thus, this ratio is not relevant.
3. [tex]\(\frac{RX}{XT}\)[/tex]
Let's examine this option:
- If we consider [tex]\(RX\)[/tex] as a side of [tex]\(\triangle RYX\)[/tex] that corresponds to [tex]\(RT\)[/tex] of [tex]\(\triangle RST\)[/tex], then this option suggests looking at the ratio by switching the positions of [tex]\(RX\)[/tex] in the numerator, aligning the comparison.
- Given that both triangles are similar by SSS theorem, [tex]\( \frac{RT}{RX} = \frac{RX}{XT} \)[/tex] by proportionality of corresponding sides given [tex]\(\triangle RYX \sim \triangle RST\)[/tex].
Therefore, this option makes sense as it compares corresponding segments under similarity transformation.
4. [tex]\(\frac{ST}{VX}\)[/tex]
This ratio involves sides [tex]\(ST\)[/tex] and [tex]\(VX\)[/tex]. The side [tex]\(VX\)[/tex] is not defined in the context of the problem, therefore this ratio does not relate to any mentioned correspondence between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
Thus, after examining all the options,
the correct ratio is:
[tex]\[ \frac{RX}{XT} \][/tex]
The corresponding ratio equivalent to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is:
[tex]\[ \boxed{\frac{RX}{XT}} \][/tex]
Given two triangles, [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], we are told they are similar. This means:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} \][/tex]
We need to determine which of the given ratios are equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Let's go through each of the options:
1. [tex]\(\frac{XY}{TS}\)[/tex]
To be a valid ratio for similar triangles, the sides in this ratio must be corresponding sides of the similar triangles. The sides [tex]\(XY\)[/tex] and [tex]\(TS\)[/tex] are not corresponding sides between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex]. Therefore, this ratio is not equivalent to [tex]\(\frac{RT}{RX}\)[/tex].
2. [tex]\(\frac{SY}{RY}\)[/tex]
Here, the side [tex]\(SY\)[/tex] is involved, but it is not mentioned which triangle's corresponding side ratio it is equivalent to. Additionally, [tex]\(RY\)[/tex] is one of the original sides given, and it does not establish the necessary proportionality with [tex]\(RT\)[/tex] or [tex]\(RX\)[/tex]. Thus, this ratio is not relevant.
3. [tex]\(\frac{RX}{XT}\)[/tex]
Let's examine this option:
- If we consider [tex]\(RX\)[/tex] as a side of [tex]\(\triangle RYX\)[/tex] that corresponds to [tex]\(RT\)[/tex] of [tex]\(\triangle RST\)[/tex], then this option suggests looking at the ratio by switching the positions of [tex]\(RX\)[/tex] in the numerator, aligning the comparison.
- Given that both triangles are similar by SSS theorem, [tex]\( \frac{RT}{RX} = \frac{RX}{XT} \)[/tex] by proportionality of corresponding sides given [tex]\(\triangle RYX \sim \triangle RST\)[/tex].
Therefore, this option makes sense as it compares corresponding segments under similarity transformation.
4. [tex]\(\frac{ST}{VX}\)[/tex]
This ratio involves sides [tex]\(ST\)[/tex] and [tex]\(VX\)[/tex]. The side [tex]\(VX\)[/tex] is not defined in the context of the problem, therefore this ratio does not relate to any mentioned correspondence between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
Thus, after examining all the options,
the correct ratio is:
[tex]\[ \frac{RX}{XT} \][/tex]
The corresponding ratio equivalent to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is:
[tex]\[ \boxed{\frac{RX}{XT}} \][/tex]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.
