Solving A Trapezoid Geometry Problem: Step-by-Step Guide

Hey guys! Let's dive into a fun geometry problem involving a trapezoid. We're given a quadrilateral $ABCD$, which is a right trapezoid. This means that sides $BC$ and $AD$ are parallel ($BC ext{ ∥ } AD$), and side $AB$ is perpendicular to side $AD$ (AB ot AD). We also have a segment $CM$, which is the height of the trapezoid. We know that angle $D$ is $45^ ext{o}$, the length of $AB$ is $5$ cm, and the length of $AD$ is $9$ cm. Our goal is to find the missing lengths. Let's break this down step-by-step. Understanding the properties of trapezoids and right angles is super important here, so make sure you're comfortable with those concepts before we jump in.

Before we begin, remember the essential properties of a trapezoid: It's a quadrilateral with at least one pair of parallel sides. In a right trapezoid, one of the non-parallel sides is perpendicular to the bases (the parallel sides). This creates two right angles at the vertices where the perpendicular side meets the bases. This problem is really about applying geometric principles and utilizing the given information effectively. We need to identify relevant triangles, and then use the properties of those triangles and angles to find unknown lengths. So, grab your pencils and let's start solving this geometry problem together! Ready? Let's go!

Finding the Height: $CM$

First things first, we need to find the length of $CM$. Since $CM$ is the height of the trapezoid and $AB$ is also the height, we know that $CM$ is equal to $AB$. Guys, this is because $AB$ and $CM$ are both perpendicular to $AD$ and $BC$, and the distance between the parallel lines $AD$ and $BC$ remains constant. Therefore, we can immediately state that: $CM = AB$. We are given that $AB = 5 ext{ cm}$. So, $CM = 5 ext{ cm}$. Boom! We've found our first missing value. Keep in mind that understanding the properties of geometric shapes, such as trapezoids, is extremely important for solving these types of problems. Visualizing the problem can also be helpful. It's often useful to sketch the trapezoid and label all the given information. Also, drawing auxiliary lines, such as the height $CM$, can often reveal useful relationships and help to form triangles or other geometric figures that can assist with solving the problem. So always take the time to create a visual representation of your problem, as it will likely make the solution clearer. Always double-check your work, particularly when dealing with geometric measurements and calculations. Make sure that all the units are consistent and that the final answers make sense in the context of the problem. This will help you catch any errors or miscalculations.

Calculation Steps for $CM$

• $CM$ is the height of the trapezoid.
• $AB$ is also the height of the trapezoid.
• Since the height is consistent across the trapezoid, $CM = AB$.
• Given $AB = 5 ext{ cm}$, therefore $CM = 5 ext{ cm}$.

Determining the Length: $MD$

Next up, we need to determine the length of $MD$. To do this, we'll use our knowledge of right triangles and the given angle. We know that $\angle D = 45^ ext{o}$. Let's consider the right triangle $\triangle CMD$. This is a right triangle because $CM$ is perpendicular to $AD$. Since $\angle D = 45^ ext{o}$, and the sum of angles in a triangle is $180^ ext{o}$, the other angle in the triangle must also be $45^ ext{o}$. This means that $\triangle CMD$ is a $45-45-90$ right triangle. What's so special about a $45-45-90$ triangle? Well, it tells us that the two legs of the triangle are equal in length. Therefore, $CM = MD$. We already know that $CM = 5 ext{ cm}$, so we can conclude that $MD = 5 ext{ cm}$. Easy, right? Remember, with these geometry problems, visualizing and identifying the right triangles within the trapezoid is key. Also, knowing your basic geometric rules like the properties of $45-45-90$ triangles, helps immensely. We're using the properties of special right triangles. A $45-45-90$ triangle has two equal sides and the hypotenuse is $\sqrt{2}$ times the length of either leg. We have our legs as $CM$ and $MD$, which are equal, and the hypotenuse is $CD$. Therefore, we have successfully calculated $MD$ using the properties of a special right triangle.

Calculation Steps for $MD$

• Identify $\triangle CMD$ as a right triangle.
• $\angle D = 45^ ext{o}$, making it a $45-45-90$ triangle.
• In a $45-45-90$ triangle, the legs are equal, so $CM = MD$.
• Since $CM = 5 ext{ cm}$, then $MD = 5 ext{ cm}$.

Finding the Length: $BC$

Finally, we need to find the length of $BC$. To do this, we first need to determine the length of $AM$. Since $CM$ is perpendicular to $AD$, $AMCD$ forms a rectangle. We already know that $AD = 9 ext{ cm}$ and $MD = 5 ext{ cm}$. $AM$ can be found by subtracting $MD$ from $AD$. This means, $AM = AD - MD = 9 ext{ cm} - 5 ext{ cm} = 4 ext{ cm}$. However, $BC$ is a bit more complex, because we need to understand how the sides are related. In a right trapezoid, $BC$ is the same length as $AM$. Therefore, $BC = AM$. We have found $AM = 4 ext{ cm}$, which means $BC = 4 ext{ cm}$. This is how we find $BC$. We can use our findings in the problem. The best thing is to break down the problem into smaller, more manageable steps. By understanding and applying the properties of trapezoids, right angles, and special triangles, we have successfully found all the missing values. It's often helpful to write down the properties and formulas that you are using. This helps in understanding the steps. Keep practicing, and you will become a geometry pro in no time! Also, review the solutions and understand the logic behind each step. This way, you are not just solving problems, but also learning important concepts that are essential in geometry. Practice makes perfect. So, keep practicing these types of problems to enhance your problem-solving skills.

Calculation Steps for $BC$

• $AM = AD - MD$.
• $AD = 9 ext{ cm}$ and $MD = 5 ext{ cm}$, so $AM = 9 ext{ cm} - 5 ext{ cm} = 4 ext{ cm}$.
• $BC = AM$.
• Therefore, $BC = 4 ext{ cm}$.

Summary of Results

Alright, guys, let's recap our findings:

1. $CM = 5 ext{ cm}$
2. $MD = 5 ext{ cm}$
3. $BC = 4 ext{ cm}$

We did it! We successfully solved the trapezoid geometry problem. You can see how breaking down the problem into smaller steps and using the properties of shapes made it easy to get to the answer. Keep practicing, and you'll become a geometry whiz in no time. If you have any questions, feel free to ask. Cheers!