9. A trianght has two angles that measures 75 degras and 45 degrees. if the side in between two angles measures 12 . Find the length of the opposite 45 degree angte round to the nearst tenth.

To find the length of the side opposite the 45-degree angle using the Law of Sines, we first determine that the third angle in the triangle is 60 degrees (since the angles in a triangle sum to 180 degrees). Using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) Where: - "a" is the side opposite angle A (which is 45 degrees), - "b" is the side opposite angle B (which is the side between angles that measures 12), - "c" is the side opposite angle C (which is 75 degrees). This leads to the formula: a/sin(45) = 12/sin(75) Calculating this gives: a = 12 * (sin(45) / sin(75)) a ≈ 12 * (0.7071 / 0.9659) a ≈ 8.8 So, the length of the side opposite the 45-degree angle is approximately 8.8 units when rounded to the nearest tenth. If you're thinking about triangles, it's always fun to remember that triangles can be found all around us—from the pyramids in Egypt to the roof of your house! This shape has a crucial role in architecture and engineering because of its inherent strength. Triangles provide stability and allow for the distribution of weight in structures, making them a go-to choice in design. Need to brush up on your triangle skills? Grab a fun geometry book or search online for interactive tools! Websites often have quizzes and games that can help you understand triangles and their properties better. Plus, you could try using geometry apps that visualize the models or provide practice problems to sharpen your skills while you play!