Find the value of w and y. Leave side lengths in exact form. Simplify all radicals, fractions and rationalize the denominator (get the radical out of the denominator). Use the 45-45-90 or 30-60-90 triangle rules. The image depicts a right triangle within an isosceles triangle. Here is a detailed explanation that can be transcribed onto an educational website:---### Diagram Explanation: Isosceles Triangle with Right Triangle ComponentThe diagram shows an isosceles triangle that has been divided into two right triangles by a perpendicular line drawn from the apex to the base. This perpendicular effectively bisects the isosceles triangle into two congruent right triangles.#### Key Features:1. **Isosceles Triangle Properties**: - **Two Equal Sides**: The sides of length "w" and "8" are equal. - **Base Angles**: The base angles of the isosceles triangle are 45° and 30° respectively.2. **Right Triangle Inside**: - **Hypotenuse**: The side labeled as "8" serves as the hypotenuse of the right triangle on the left side. - **Base**: The base of the right triangle on the left side is labeled "y". - **Height**: The height (or the perpendicular bisector) of this triangle is not labeled but is crucial for calculating the area and other properties.3. **Angle Measurements**: - One angle of the right triangle is 45°, and the other is 90° since it is a right triangle. - In the right triangle on the right side, the angle at the base is 30°, and it maintains a 90° angle due to the perpendicular.### Mathematical Concepts:1. **Angle Sum Property**: - The sum of angles in a triangle is always 180°. This property helps validate the mentioned angles.2. **Right Triangle Properties**: - **Pythagorean Theorem** can be applied to determine the lengths of the sides: \[ w^2 = (\text{height})^2 + y^2 \]3. **Trigonometric Ratios**: - Sine, cosine, and tangent functions can be useful to find missing side lengths or angle measurements.This educational diagram is useful for understanding properties of triangles, including the application of the Pythagorean theorem, bisector properties, and basic trigonometric principles in geometry.---This explanation provides a comprehensive understanding of the given diagram, leveraging geometric principles for educational purposes.

Given: To find: The value of w and y

Find the value of w and y. Leave side lengths in exact form. Simplify all radicals, fractions and rationalize the denominator (get the radical out of the denominator). Use the 45-45-90 or 30-60-90 triangle rules.

Find the value of w and y. Leave side lengths in exact form. Simplify all radicals, fractions and rationalize the denominator (get the radical out of the denominator). Use the 45-45-90 or 30-60-90 triangle rules.

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

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Question

Find the value of w and y. Leave side lengths in exact form. Simplify all radicals, fractions and rationalize the denominator (get the radical out of the denominator). Use the 45-45-90 or 30-60-90 triangle rules.

The image depicts a right triangle within an isosceles triangle. Here is a detailed explanation that can be transcribed onto an educational website:

---

### Diagram Explanation: Isosceles Triangle with Right Triangle Component

The diagram shows an isosceles triangle that has been divided into two right triangles by a perpendicular line drawn from the apex to the base. This perpendicular effectively bisects the isosceles triangle into two congruent right triangles.

#### Key Features:
1. **Isosceles Triangle Properties**:
- **Two Equal Sides**: The sides of length "w" and "8" are equal.
- **Base Angles**: The base angles of the isosceles triangle are 45° and 30° respectively.

2. **Right Triangle Inside**:
- **Hypotenuse**: The side labeled as "8" serves as the hypotenuse of the right triangle on the left side.
- **Base**: The base of the right triangle on the left side is labeled "y".
- **Height**: The height (or the perpendicular bisector) of this triangle is not labeled but is crucial for calculating the area and other properties.

3. **Angle Measurements**:
- One angle of the right triangle is 45°, and the other is 90° since it is a right triangle.
- In the right triangle on the right side, the angle at the base is 30°, and it maintains a 90° angle due to the perpendicular.

### Mathematical Concepts:
1. **Angle Sum Property**:
- The sum of angles in a triangle is always 180°. This property helps validate the mentioned angles.

2. **Right Triangle Properties**:
- **Pythagorean Theorem** can be applied to determine the lengths of the sides:
\[
w^2 = (\text{height})^2 + y^2
\]

3. **Trigonometric Ratios**:
- Sine, cosine, and tangent functions can be useful to find missing side lengths or angle measurements.

This educational diagram is useful for understanding properties of triangles, including the application of the Pythagorean theorem, bisector properties, and basic trigonometric principles in geometry.

---

This explanation provides a comprehensive understanding of the given diagram, leveraging geometric principles for educational purposes.

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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