https://file.aiquickdraw.com/mathbot/file/17303861182347cxan648.jpg

To solve this right triangle, we’ll follow these steps to determine the values of \( XY \), \( \angle X \), and \( \angle Z \):

1. **Identify the triangle sides and angles:**
   - Given \( Y \) is a right angle (90°).
   - \( XY \) is the unknown leg (opposite of \(\angle Z\)).
   - \( YZ = 14 \) (opposite of \(\angle X\)).
   - \( XZ = 19 \) (hypotenuse).

2. **Calculate \( XY \) using the Pythagorean theorem:**
   \[
   XY = \sqrt{XZ^2 - YZ^2} = \sqrt{19^2 - 14^2} = \sqrt{361 - 196} = \sqrt{165} \approx 13
   \]

3. **Calculate \( \angle X \) using trigonometry (sine):**
   \[
   \sin(\angle X) = \frac{YZ}{XZ} = \frac{14}{19} \approx 0.7368
   \]
   \[
   \angle X \approx \arcsin(0.7368) \approx 48^\circ
   \]

4. **Calculate \( \angle Z \) since it complements \( \angle X \):**
   \[
   \angle Z = 90^\circ - \angle X \approx 90^\circ - 48^\circ = 42^\circ
   \]

### Final Answers:
- \( XY \approx 13 \)
- \( m\angle X \approx 48^\circ \)
- \( m\angle Z \approx 42^\circ \)

The lock code (in order of answers) is: **13, 48, 42**

Would you like further details or have questions about this solution?

---

1. How is the Pythagorean theorem used to solve for side lengths?
2. What are the main trigonometric ratios in a right triangle?
3. How can you use complementary angles in right triangles?
4. Why is rounding necessary in trigonometric calculations?
5. How would the process change if \( \angle X \) were given instead?

**Tip:** Remember that in a right triangle, the two acute angles add up to 90°.

Solve this triangle. Given that triangle XYZ is a right triangle with sides YZ = 14 and XZ = 19, find the length of XY and the measures of angles X and Z. Round all answers to the nearest whole number.

This guide explains how to solve a right triangle with side lengths YZ = 14 and XZ = 19 by finding the unknown side length XY and the angle measures of X and Z. Using the Pythagorean theorem and trigonometric functions, this problem is suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation