Is Quadrilateral ABCD A Trapezoid? A Step-by-Step Guide

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Is Quadrilateral ABCD a Trapezoid? A Step-by-Step Guide

Hey guys! Let's dive into a geometry problem and figure out if a quadrilateral is a trapezoid. We'll be working with the quadrilateral ABCD, which has vertices at A(-4, -5), B(-3, 0), C(0, 2), and D(5, 1). Remember, a trapezoid is a quadrilateral with at least one pair of parallel sides. Our mission? To find out if ABCD fits the bill. This involves calculating the slopes of its sides and seeing if any pairs are equal. Let's break it down step-by-step to make sure we do it right. Ready to get started?

Step 1: Finding the Slope of AB

Alright, first things first: we need to find the slope of line segment AB. The slope of a line is a measure of its steepness and direction. It's calculated using the formula: slope = (y₂ - y₁) / (x₂ - x₁). In our case, A is (-4, -5) and B is (-3, 0). So, let's plug in those coordinates into the slope formula. This is the critical first step. Make sure you don't mess it up! Substituting the coordinates of A and B, we get:

  • slope of AB = (0 - (-5)) / (-3 - (-4))

Let's simplify that:

  • slope of AB = (0 + 5) / (-3 + 4) = 5/1 = 5

So, the slope of AB is 5. This tells us how steeply the line AB rises (or falls) as we move from left to right. A positive slope, like in this case, means the line is going uphill as you move from left to right. Easy peasy, right? We've knocked out the first step, and we're one step closer to figuring out if ABCD is a trapezoid. Good job! Keep it up.

Now, let's remember why we're doing all this. The slope helps us determine if sides are parallel. Parallel lines have the same slope. If we find another side with a slope of 5, then we know we've got a pair of parallel lines. Let's move on to the next step and see what we get.

To ensure we are on the right track, let's take a look back at our work. We carefully used the formula to calculate the slope using the coordinates of A and B. We correctly subtracted the y-coordinates and the x-coordinates, paying close attention to the negative signs. The result of 5/1 or 5 seems reasonable, and it gives us the direction and steepness of the line AB. Now, let's move forward.

Step 2: Finding the Slope of DC

Okay, time for the second step! We're now going to calculate the slope of the line segment DC. We'll use the same slope formula: slope = (y₂ - y₁) / (x₂ - x₁). This time, we're using the coordinates of D (5, 1) and C (0, 2). Let's plug those values into the formula:

  • slope of DC = (2 - 1) / (0 - 5)

Now, let's simplify:

  • slope of DC = 1 / -5 = -1/5

So, the slope of DC is -1/5. Notice that this slope is negative, which means the line DC goes downhill as you move from left to right. Now we have two slopes: the slope of AB is 5, and the slope of DC is -1/5. Are these lines parallel? Nope! They have different slopes, so AB and DC are not parallel. We'll keep this in mind. Remember, for a shape to be a trapezoid, it only needs one pair of parallel sides. So, we're not done yet, guys! Let's continue on to the other sides.

Remember, we are looking for parallel lines to determine if the shape is a trapezoid. So, we are going to look for another pair of sides that have the same slope. Since the slope of AB is 5 and the slope of DC is -1/5, it seems that those sides are not parallel. We will continue checking the remaining sides of the quadrilateral ABCD. We have calculated the slope of AB and DC. It is time to determine the slopes of the other sides, i.e., BC and AD. Let's do it!

To make sure we're on the right track, let's revisit our work. We plugged in the coordinates of D and C into the slope formula, subtracted correctly, and arrived at a result of -1/5. This negative slope confirms that DC slopes downwards as it moves from left to right. Also, since we got two different slopes for AB and DC, we know that these two lines are not parallel.

Step 3: Finding the Slope of BC

Let's get the slope of BC. We'll use the trusty slope formula: slope = (y₂ - y₁) / (x₂ - x₁). This time, we use the coordinates of B (-3, 0) and C (0, 2). Substituting these values into the formula, we get:

  • slope of BC = (2 - 0) / (0 - (-3))

Simplify the expression:

  • slope of BC = 2 / (0 + 3) = 2/3

So, the slope of BC is 2/3. Now we have three slopes: AB (5), DC (-1/5), and BC (2/3). None of these slopes are the same, so no pair of lines are parallel, among AB, DC, and BC. We're still trying to determine if quadrilateral ABCD is a trapezoid, so we need to find the slope of the last side, which is AD. Let's get to it!

As you see, the slope of BC is 2/3. Let's check our work. We correctly substituted the x and y coordinates, carefully handling the negative signs, and got the result 2/3. We have yet to find parallel sides, but we're getting closer to making a conclusion. We've got the slopes for AB, DC, and BC. Next, we are going to check the slope of the last side, AD. Let's keep going.

Step 4: Finding the Slope of AD

Alright, last one! Let's find the slope of AD. The formula is the same: slope = (y₂ - y₁) / (x₂ - x₁). This time, use the coordinates of A (-4, -5) and D (5, 1). Plug the values into the formula:

  • slope of AD = (1 - (-5)) / (5 - (-4))

Now, let's simplify:

  • slope of AD = (1 + 5) / (5 + 4) = 6/9 = 2/3

So, the slope of AD is 2/3. This is the same slope we got for BC! This is the breakthrough moment, guys! Since BC and AD have the same slope (2/3), they are parallel. Remember, a trapezoid needs one pair of parallel sides. We've found it!

Step 5: Conclusion

Conclusion: Quadrilateral ABCD is a trapezoid because it has at least one pair of parallel sides (BC and AD). That's all there is to it! We found our answer by carefully calculating the slopes of all four sides and checking for any pairs that were equal. Great job, everyone!

To recap:

  • Slope of AB = 5
  • Slope of DC = -1/5
  • Slope of BC = 2/3
  • Slope of AD = 2/3

Since BC and AD have the same slope, they are parallel. Therefore, ABCD is a trapezoid. We used the slope formula to identify these parallel sides, which is all we needed to determine the nature of the shape. We can confidently say that the given shape is a trapezoid because it satisfies the required definition. Well done!

Also, remember that even if other sides weren't parallel, as long as one pair is, it is a trapezoid. This is why it is important to check all the pairs of opposite sides. It makes sure that we identify the pairs of parallel lines, and in the end, it tells us if the quadrilateral is a trapezoid. Hope this helped you to understand better how to calculate the slope and identify trapezoids. If you have any questions, don't hesitate to ask! Thanks for following along, and keep practicing your geometry skills! You've got this!