The image shows two right-angled triangles with some given angles and side lengths.

Analyzing the Triangles:

1. Left Triangle:

   • It is a right-angled triangle.
   • Given angles: $45^\circ$ and $45^\circ$, indicating this is an isosceles right triangle.
   • Given side: One side (the leg opposite one of the $45^\circ$ angles) is marked as 4 units.

2. Right Triangle:

   • This is also a right-angled triangle.
   • Given angles: $30^\circ$, $60^\circ$, and the right angle $90^\circ$.
   • Given side: The side opposite the $30^\circ$ angle is marked as 6 units.

Solving for the Unknown Sides:

Let's solve for the missing sides in both triangles:

1. Left Triangle:

   • Since it is a 45-45-90 triangle, both legs are equal. So, if one leg is 4, the other leg is also 4.
   • In a 45-45-90 triangle, the hypotenuse is $\sqrt{2}$ times the length of a leg. $\text{Hypotenuse} = 4 \cdot \sqrt{2} \approx 5.66$

2. Right Triangle:

   • In a 30-60-90 triangle, the ratio of the sides opposite the $30^\circ$, $60^\circ$, and $90^\circ$ angles is $1 : \sqrt{3} : 2$.
   • Given that the side opposite the $30^\circ$ angle is 6, the hypotenuse (opposite the $90^\circ$ angle) will be: $\text{Hypotenuse} = 2 \times 6 = 12$
   • The side opposite the $60^\circ$ angle will be: $\text{Side opposite } 60^\circ = 6 \cdot \sqrt{3} \approx 10.39$

Would you like to know more details or ask any questions?

Related Questions:

1. How do you derive the side ratios for 30-60-90 and 45-45-90 triangles?
2. What are the properties of isosceles right triangles?
3. How can trigonometry be used to find unknown sides in right triangles?
4. What are some real-life applications of special right triangles?
5. How do you find the area of these triangles?

Tip:

When working with right triangles, remember that the Pythagorean theorem and trigonometric ratios can help solve for unknown sides or angles efficiently!