On an educational website, the transcription and explanation of the image would be as follows: --- ### Understanding Tangent Ratios in Right Triangles In the study of trigonometry, particularly with right triangles, the tangent function (tan) is an essential trigonometric ratio. Below are two right-angled triangles with one of the acute angles labeled as \(\alpha\): #### Triangle 1 - Hypotenuse: \(z\) - Opposite side (to angle \(\alpha\)): \(x\) - Adjacent side (to angle \(\alpha\)): \(y\) #### Triangle 2 - Hypotenuse: \(s\) - Opposite side (to angle \(\alpha\)): \(q\) - Adjacent side (to angle \(\alpha\)): \(r\) The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as: \[ \tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}} \] ### Question c. Which ratios of side lengths are equal to \(\tan(\alpha)\)? Select all that apply: - \(\frac{r}{q}\) - \(\frac{y}{z}\) - \(\frac{z}{x}\) - \(\frac{r}{s}\) - \(\frac{y}{x}\) ### Explanation To determine the correct ratios, we need to identify the ratios where the opposite side to \(\alpha\) is divided by the adjacent side of \(\alpha\): #### For Triangle 1: - \(\tan(\alpha) = \frac{x}{y}\) #### For Triangle 2: - \(\tan(\alpha) = \frac{q}{r}\) Thus, the ratios \(\frac{y}{x}\) and \(\frac{q}{r}\) will be equal to \(\tan(\alpha)\). The correct answers are: - \(\boxed{\frac{y}{x}}\) --- This explanation incorporates both the textual and graphical elements of the image to provide a comprehensive understanding of how to determine which ratios correspond to the tangent of angle \(\alpha\).
On an educational website, the transcription and explanation of the image would be as follows: --- ### Understanding Tangent Ratios in Right Triangles In the study of trigonometry, particularly with right triangles, the tangent function (tan) is an essential trigonometric ratio. Below are two right-angled triangles with one of the acute angles labeled as \(\alpha\): #### Triangle 1 - Hypotenuse: \(z\) - Opposite side (to angle \(\alpha\)): \(x\) - Adjacent side (to angle \(\alpha\)): \(y\) #### Triangle 2 - Hypotenuse: \(s\) - Opposite side (to angle \(\alpha\)): \(q\) - Adjacent side (to angle \(\alpha\)): \(r\) The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as: \[ \tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}} \] ### Question c. Which ratios of side lengths are equal to \(\tan(\alpha)\)? Select all that apply: - \(\frac{r}{q}\) - \(\frac{y}{z}\) - \(\frac{z}{x}\) - \(\frac{r}{s}\) - \(\frac{y}{x}\) ### Explanation To determine the correct ratios, we need to identify the ratios where the opposite side to \(\alpha\) is divided by the adjacent side of \(\alpha\): #### For Triangle 1: - \(\tan(\alpha) = \frac{x}{y}\) #### For Triangle 2: - \(\tan(\alpha) = \frac{q}{r}\) Thus, the ratios \(\frac{y}{x}\) and \(\frac{q}{r}\) will be equal to \(\tan(\alpha)\). The correct answers are: - \(\boxed{\frac{y}{x}}\) --- This explanation incorporates both the textual and graphical elements of the image to provide a comprehensive understanding of how to determine which ratios correspond to the tangent of angle \(\alpha\).
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![On an educational website, the transcription and explanation of the image would be as follows:
---
### Understanding Tangent Ratios in Right Triangles
In the study of trigonometry, particularly with right triangles, the tangent function (tan) is an essential trigonometric ratio.
Below are two right-angled triangles with one of the acute angles labeled as \(\alpha\):
#### Triangle 1
- Hypotenuse: \(z\)
- Opposite side (to angle \(\alpha\)): \(x\)
- Adjacent side (to angle \(\alpha\)): \(y\)
#### Triangle 2
- Hypotenuse: \(s\)
- Opposite side (to angle \(\alpha\)): \(q\)
- Adjacent side (to angle \(\alpha\)): \(r\)
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as:
\[
\tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}}
\]
### Question
c. Which ratios of side lengths are equal to \(\tan(\alpha)\)?
Select all that apply:
- \(\frac{r}{q}\)
- \(\frac{y}{z}\)
- \(\frac{z}{x}\)
- \(\frac{r}{s}\)
- \(\frac{y}{x}\)
### Explanation
To determine the correct ratios, we need to identify the ratios where the opposite side to \(\alpha\) is divided by the adjacent side of \(\alpha\):
#### For Triangle 1:
- \(\tan(\alpha) = \frac{x}{y}\)
#### For Triangle 2:
- \(\tan(\alpha) = \frac{q}{r}\)
Thus, the ratios \(\frac{y}{x}\) and \(\frac{q}{r}\) will be equal to \(\tan(\alpha)\).
The correct answers are:
- \(\boxed{\frac{y}{x}}\)
---
This explanation incorporates both the textual and graphical elements of the image to provide a comprehensive understanding of how to determine which ratios correspond to the tangent of angle \(\alpha\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda1f6cc5-8a6c-4e05-8a10-05c0cc87bf21%2Fccd8b120-f3e0-47b8-9dbe-4d35451f5a6e%2Fv9mdvdt.jpeg&w=3840&q=75)
Transcribed Image Text:On an educational website, the transcription and explanation of the image would be as follows:
---
### Understanding Tangent Ratios in Right Triangles
In the study of trigonometry, particularly with right triangles, the tangent function (tan) is an essential trigonometric ratio.
Below are two right-angled triangles with one of the acute angles labeled as \(\alpha\):
#### Triangle 1
- Hypotenuse: \(z\)
- Opposite side (to angle \(\alpha\)): \(x\)
- Adjacent side (to angle \(\alpha\)): \(y\)
#### Triangle 2
- Hypotenuse: \(s\)
- Opposite side (to angle \(\alpha\)): \(q\)
- Adjacent side (to angle \(\alpha\)): \(r\)
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as:
\[
\tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}}
\]
### Question
c. Which ratios of side lengths are equal to \(\tan(\alpha)\)?
Select all that apply:
- \(\frac{r}{q}\)
- \(\frac{y}{z}\)
- \(\frac{z}{x}\)
- \(\frac{r}{s}\)
- \(\frac{y}{x}\)
### Explanation
To determine the correct ratios, we need to identify the ratios where the opposite side to \(\alpha\) is divided by the adjacent side of \(\alpha\):
#### For Triangle 1:
- \(\tan(\alpha) = \frac{x}{y}\)
#### For Triangle 2:
- \(\tan(\alpha) = \frac{q}{r}\)
Thus, the ratios \(\frac{y}{x}\) and \(\frac{q}{r}\) will be equal to \(\tan(\alpha)\).
The correct answers are:
- \(\boxed{\frac{y}{x}}\)
---
This explanation incorporates both the textual and graphical elements of the image to provide a comprehensive understanding of how to determine which ratios correspond to the tangent of angle \(\alpha\).
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