Finding Diagonal AC: A Geometry Problem Solved

Hey guys! Today, we're diving into a classic geometry problem. We'll explore how to find the length of a diagonal in a rectangle when we know one side and an angle. This is a super common type of problem, so understanding it will seriously boost your geometry game. Let's break down the problem, step by step, and make sure it's crystal clear. We'll be using some basic trigonometry and the properties of rectangles to nail this one. So, grab your pencils and let's get started! This is going to be fun, I promise. Understanding the problem is the first and most important step in the solution. In the context of the geometry problem, we are given a rectangle ABCD where side AB has a length of 12 cm, and angle ABD is 60 degrees. The question asks to find the length of diagonal AC. Let's visualize this: Imagine a rectangle. Now, draw a diagonal, AC, that splits the rectangle into two right-angled triangles. The key here is that all angles in a rectangle are 90 degrees. This is our starting point. We also know properties of right angle triangles. This will be useful! This is very important because we can use properties that we know of 90 degrees angle triangles to find the diagonal AC. So, how do we solve it? Let's explore a few approaches, but we'll focus on the most direct and efficient methods. Remember, in geometry, there's often more than one way to skin a cat! Our main keywords here are: rectangle, diagonal, angle, trigonometry.

Breaking Down the Rectangle: The Power of Right Triangles

Okay, let's get down to business. The problem gives us a rectangle, and as we already know, the angles inside a rectangle are all right angles (90 degrees). Specifically, angle ABC, BCD, CDA, and DAB are right angles. Angle ABD, which is 60 degrees, is also given. Consider right triangle ABD. We know the length of AB (12 cm) and the angle ABD (60°). This gives us everything we need to find the length of BD (the other side of the triangle), and eventually, AC, the diagonal. Also, we know that the angle BAD is a right angle, which is 90 degrees. Using the fact that the sum of the angles in a triangle is always 180 degrees, we can find the angle ADB. Angle ADB = 180° - 90° - 60° = 30°. Now, we have all angles and one side of the right-angled triangle ABD. We can proceed by using some trigonometry. The tools we'll need are the sine, cosine, and tangent functions. Remember SOH CAH TOA? It's your best friend in trigonometry! In our right triangle ABD, let's figure out which trigonometric function to use. To find the length of BD, we can use the tangent function. We know the adjacent side (AB = 12 cm) and the angle ABD (60°), and we want to find the opposite side (AD). Alternatively, we can use the cosine to find the length of BD, which is the hypotenuse. Cosine (cos) relates the adjacent side (AB) to the hypotenuse (BD). So, we have cos(60°) = AB / BD, or cos(60°) = 12 cm / BD. The cool part is that this will give us the length of BD, so we can use the Pythagorean theorem to find the length of the other diagonal. Here is another important part that we need to remember. Diagonals in a rectangle are equal, meaning that AC = BD. So finding BD is the key to solve it. Now, let's figure out the cosine of 60 degrees. cos(60°) = 0.5. Now we need to rearrange the formula: 0.5 = 12 cm / BD => BD = 12 cm / 0.5 => BD = 24 cm. So, we found the length of the diagonal BD, which is 24cm. Because the diagonals are equal, then the length of AC is also 24 cm. Thus we solved the problem. The main keywords are: right triangles, trigonometry, cosine, diagonals.

Using Trigonometry to Solve the Problem

Alright, let's crank up the trigonometry a bit. We're given the right triangle ABD, with AB = 12 cm and angle ABD = 60°. We want to find the length of the diagonal AC. Since diagonals in a rectangle are equal, finding the length of BD is the same as finding the length of AC. To do this, we will use trigonometric functions to determine BD. As mentioned above, we can use cosine. cos(angle) = Adjacent / Hypotenuse. The angle ABD is 60°, the adjacent side is AB (12 cm), and the hypotenuse is BD (the diagonal we want to find). So, cos(60°) = AB / BD. Now, we know that cos(60°) = 0.5. So, 0.5 = 12 cm / BD. To solve for BD, we rearrange the equation: BD = 12 cm / 0.5, therefore BD = 24 cm. Here is the amazing part. Since ABCD is a rectangle, AC = BD. We now know that AC = 24 cm. And just like that, we found the length of the diagonal AC! Another method is using sine and tangent functions to solve the problem. We can find the length of AD using the tangent or sine function. tan(60°) = AD / AB => AD = AB * tan(60°) => AD = 12 cm * √3. Then, we can use the Pythagorean theorem to find the length of BD: BD² = AB² + AD². To find AD, we can also use sine. sin(60°) = AD / BD => AD = BD * sin(60°). So we see multiple methods of finding the same thing, and here is the power of Geometry! Key takeaways: Remember to use trigonometric functions and Pythagorean theorem for calculations. Also, don't forget the properties of rectangle and right-angle triangles. Main keywords are: trigonometry, cosine, sine, tangent.

The Pythagorean Theorem: An Alternative Route

Now, let's take a detour and solve this using the Pythagorean theorem. This is another powerful tool in geometry, and it often comes in handy. Remember that the Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our triangle ABD, BD is the hypotenuse, AB and AD are the other two sides. We know AB = 12 cm. But we need to find AD. Now we can use the properties of 30-60-90 degree triangles. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the shorter side. Let's look again at the right triangle ABD. We know that angle ABD = 60°, and angle ADB = 30°. Using that information, we know that side AD is opposite to the 60-degree angle, so it is √3 times the shorter side (which is half of BD). The length of AD is 12 * √3 cm. Now, we can use the Pythagorean theorem: BD² = AB² + AD². BD² = 12² + (12 * √3)² = 144 + 432 = 576. So, BD = √576 = 24 cm. Again, AC = BD = 24 cm. Bam! We arrived at the same answer using a different method. Pretty cool, huh? The Pythagorean theorem is a game-changer. This also helps to confirm our previous answer. Make sure you are comfortable with all these methods. Main keywords are: Pythagorean theorem, 30-60-90 triangles, square root.

Step-by-Step Solution Using the Pythagorean Theorem

Here's a clear, step-by-step breakdown to show you exactly how to use the Pythagorean theorem in this scenario. Step 1: Identify the right triangle: In rectangle ABCD, triangle ABD is a right-angled triangle. Angle DAB is 90 degrees. Step 2: Determine the known sides: We know AB = 12 cm and angle ABD = 60°. Step 3: Find the missing side (AD): We can find AD using trigonometry or the properties of a 30-60-90 triangle, as we discussed earlier. Using the properties, we can find the length of AD as 12 * √3. Step 4: Apply the Pythagorean theorem: BD² = AB² + AD². BD² = 12² + (12 * √3)². BD² = 144 + 432 = 576. Step 5: Calculate the diagonal: BD = √576 = 24 cm. Step 6: The final answer: AC = BD = 24 cm. See? It's all connected. This shows you how different concepts in geometry work together. Main keywords are: step-by-step, pythagorean theorem, rectangle, triangle.

Conclusion: Bringing it All Together

So there you have it! We found the length of diagonal AC in a rectangle using trigonometry and the Pythagorean theorem. By breaking down the problem into manageable steps, and understanding the properties of rectangles and right triangles, we were able to crack the code. This problem highlights the power of understanding basic geometric principles and how they relate to each other. Remember, practice makes perfect! The more you work through these kinds of problems, the more comfortable and confident you'll become. Keep exploring, keep practicing, and never be afraid to ask questions. You got this, guys! Always remember the core concepts: understand the shape, identify the knowns, choose the right tools (trigonometry, Pythagorean theorem), and solve it. This is useful not only for geometry, but also for other problem-solving exercises. Main keywords are: conclusion, problem-solving, geometry, practice.