https://file.aiquickdraw.com/mathbot/file/1727065734743c1eky8qi.jpg

The image shows two right triangles, one with a \( 45^\circ \) angle and the other with a \( 30^\circ \) angle. The problem asks for the exact values of the sides \( b \) and \( c \), and requires answers in simplified radical form.

### First Triangle:
- This is a **45-45-90 triangle**, which has side ratios of:
  \[
  1 : 1 : \sqrt{2}
  \]
- Given that one leg is \( 8 \), the hypotenuse \( c \) will be:
  \[
  c = 8\sqrt{2}
  \]
  Since both legs are equal in a 45-45-90 triangle, \( b = 8 \).

### Second Triangle:
- This is a **30-60-90 triangle**, which has side ratios of:
  \[
  1 : \sqrt{3} : 2
  \]
- Given that the shortest side (opposite the \( 30^\circ \) angle) is \( 6 \), the hypotenuse \( c \) will be:
  \[
  c = 2 \times 6 = 12
  \]
  The longer leg \( b \) (opposite the \( 60^\circ \) angle) will be:
  \[
  b = 6\sqrt{3}
  \]

### Answers:
1. For the first triangle:
   - \( b = 8 \)
   - \( c = 8\sqrt{2} \)

2. For the second triangle:
   - \( b = 6\sqrt{3} \)
   - \( c = 12 \)

Would you like further details or explanations?

---

Here are 5 related questions to expand on this topic:
1. How are the side ratios of special right triangles derived?
2. What is the general formula for the hypotenuse in a 45-45-90 triangle?
3. Can you solve a right triangle that does not have angles of 30, 60, or 45 degrees?
4. What are the steps to simplify radical expressions in triangle problems?
5. How do you calculate the height of an equilateral triangle using these ratios?

**Tip:** In any triangle, the sum of the angles always equals 180°. This is helpful for identifying missing angles.

En los siguientes triángulos rectángulos, hallar las longitudes exactas de los lados b y c. De ser necesario, escribir las respuestas en forma radical simplificada.

Learn how to solve special right triangles, including 45°-45°-90° and 30°-60°-90° triangles. Find the exact lengths of the sides b and c, and understand how to express answers in simplified radical form. 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