All triangles are similar, solve for x, y and z as appropriate. Round any decimal answers to the nearest 10th. ### Geometric Diagram Analysis: Proportional TrianglesIn this diagram, we observe two right-angled triangles, sharing a common vertex T:1. The larger triangle is labeled TSU.2. The smaller triangle is labeled TCB.Let's break down the components of the diagram:#### Points and Segments- **T** and **U** are on the horizontal axis with TU being divided into two segments, TC and CU.- **TS** is vertically aligned at point T, extending upward to point S.- Segment **TS** is intersected by point X at an indicated distance of 12 units above point B.- **B** is directly above point U at the same height as X.- **TS** intersects point C at a distance of 18 units from point T.- **CU** measures 6 units.#### Interpretation##### Horizontal Segments:- **TU** is the base of the bigger triangle and is divided as follows: - **TC** measures 18 units. - **CU** measures 6 units.##### Vertical Segment:- The vertical distance from **B** to **U**, as indicated in the diagram, is 12 units.##### Triangle Proportions:- Triangles **TSU** and **TCB** are similar by AA (Angle-Angle) similarity criterion because each pair shares a corresponding angle at point T and the angles at points S and B are right angles. ##### Concluding Thought:Understanding the relationships between the side lengths and angles can be pivotal in solving geometry problems involving similar triangles. The proportionality allows for determining unknown lengths and verifying geometric properties. This diagram acts as an excellent example of similar triangle properties and the concept of proportional segments in right-angled triangles. ### Introduction to Proportional and Similar TrianglesUnderstanding the concepts of proportional and similar triangles is crucial in geometry. This topic helps us comprehend the relationship between similar-shaped geometric figures and how their sides and angles correspond. Below, we have an image that exhibits two triangles with their side lengths provided.#### Explanation of DiagramThe image contains two triangles. - **First Triangle** (Left): - One side of the triangle measures 63 units. - The other two sides each measure 49 units. - **Second Triangle** (Right): - One side of the triangle measures $8x - 2$ units. - Another side measures 42 units. #### Analysis:By analyzing these triangles' side lengths, particularly the corresponding side lengths between the two triangles, one can determine if they are similar (having the same shape but not necessarily the same size).For similarity:- The ratios of the lengths of corresponding sides should be equal. This means the ratios of the triangles’ sides can be compared and, if the ratios are equivalent, the triangles are similar.Let's set up the ratios with the provided lengths:- Ratio of side 63 (first triangle) to side $8x - 2$ (second triangle).- Ratio of side 49 (first triangle) to side 42 (second triangle). Using the principles of proportionality and solving these ratios will help determine the unknown variable $ x $ or confirm similarity.#### Conclusion:Understanding these relationships not only helps in solving geometric problems but also builds a foundation for more complex mathematical concepts. Practicing with different triangles, identifying corresponding sides and setting up proportional ratios are excellent ways to grasp the essence of similar triangles.

HINT: We know that if 2 triangles are similar then ratios of length of their…

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B 12 ア 18 6.

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

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Question

All triangles are similar, solve for x, y and z as appropriate.

Round any decimal answers to the nearest 10th.

### Geometric Diagram Analysis: Proportional Triangles

In this diagram, we observe two right-angled triangles, sharing a common vertex T:

1. The larger triangle is labeled TSU.
2. The smaller triangle is labeled TCB.

Let's break down the components of the diagram:

#### Points and Segments
- **T** and **U** are on the horizontal axis with TU being divided into two segments, TC and CU.
- **TS** is vertically aligned at point T, extending upward to point S.
- Segment **TS** is intersected by point X at an indicated distance of 12 units above point B.
- **B** is directly above point U at the same height as X.
- **TS** intersects point C at a distance of 18 units from point T.
- **CU** measures 6 units.

#### Interpretation

##### Horizontal Segments:
- **TU** is the base of the bigger triangle and is divided as follows:
- **TC** measures 18 units.
- **CU** measures 6 units.

##### Vertical Segment:
- The vertical distance from **B** to **U**, as indicated in the diagram, is 12 units.

##### Triangle Proportions:
- Triangles **TSU** and **TCB** are similar by AA (Angle-Angle) similarity criterion because each pair shares a corresponding angle at point T and the angles at points S and B are right angles.

##### Concluding Thought:
Understanding the relationships between the side lengths and angles can be pivotal in solving geometry problems involving similar triangles. The proportionality allows for determining unknown lengths and verifying geometric properties. This diagram acts as an excellent example of similar triangle properties and the concept of proportional segments in right-angled triangles.

### Introduction to Proportional and Similar Triangles

Understanding the concepts of proportional and similar triangles is crucial in geometry. This topic helps us comprehend the relationship between similar-shaped geometric figures and how their sides and angles correspond. Below, we have an image that exhibits two triangles with their side lengths provided.

#### Explanation of Diagram
The image contains two triangles.

- **First Triangle** (Left):
- One side of the triangle measures 63 units.
- The other two sides each measure 49 units.

- **Second Triangle** (Right):
- One side of the triangle measures $8x - 2$ units.
- Another side measures 42 units.

#### Analysis:
By analyzing these triangles' side lengths, particularly the corresponding side lengths between the two triangles, one can determine if they are similar (having the same shape but not necessarily the same size).

For similarity:
- The ratios of the lengths of corresponding sides should be equal. This means the ratios of the triangles’ sides can be compared and, if the ratios are equivalent, the triangles are similar.

Let's set up the ratios with the provided lengths:
- Ratio of side 63 (first triangle) to side $8x - 2$ (second triangle).
- Ratio of side 49 (first triangle) to side 42 (second triangle).

Using the principles of proportionality and solving these ratios will help determine the unknown variable $ x $ or confirm similarity.

#### Conclusion:
Understanding these relationships not only helps in solving geometric problems but also builds a foundation for more complex mathematical concepts. Practicing with different triangles, identifying corresponding sides and setting up proportional ratios are excellent ways to grasp the essence of similar triangles.

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

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