Is Triangle DEF A Right Triangle? The Key Condition

by Andrew McMorgan 52 views

Hey guys! Today, we're diving into the coordinate plane to check out a triangle formed by three points: D(-2, 2), E(2, 5), and F(5, 1). We're on a mission to figure out what condition tells us if this triangle is a right triangle. It's a classic geometry problem, and understanding this will seriously boost your math game.

So, we've got our points: D at (-2, 2), E at (2, 5), and F at (5, 1). To determine if triangle DEF is a right triangle, we need to recall a fundamental property of right triangles. This property relates to the slopes of the lines that form the sides of the triangle. Remember that two lines are perpendicular if, and only if, the product of their slopes is -1. This is the golden rule we'll be using, guys. It's super important to remember this relationship because it's the key to unlocking whether our triangle has that crucial 90-degree angle. We're not just guessing here; we're using a proven mathematical principle. The coordinate plane gives us the tools to calculate these slopes, and once we have them, we can apply this perpendicularity test.

Let's break down how we find the slope of a line segment between two points (x1, y1) and (x2, y2). The formula is pretty straightforward: slope (m) = (y2 - y1) / (x2 - x1). Keep this formula handy because we'll be using it for each side of our triangle DEF. It's essential to be careful with your calculations, especially when dealing with negative numbers. A small slip-up can lead to the wrong conclusion, so double-check each step. We're going to calculate the slope for the segment DE, then for EF, and finally for FD. Once we have all three slopes, the real detective work begins. We'll be looking for a pair of slopes whose product is exactly -1. If we find such a pair, boom! We've confirmed that two sides of the triangle are perpendicular, making it a right triangle. If none of the pairs multiply to -1, then it's not a right triangle, simple as that.

Understanding Slopes and Perpendicularity

Alright, let's get down to business and calculate the slopes for each side of our triangle DEF. First up, we have the segment connecting point D(-2, 2) and point E(2, 5). Using our slope formula, m = (y2 - y1) / (x2 - x1):

  • Slope of DE (m_DE) = (5 - 2) / (2 - (-2)) = 3 / (2 + 2) = 3 / 4.

So, the slope of the side DE is 3/4. Keep that number in mind. Now, let's move on to the segment connecting E(2, 5) and F(5, 1).

  • Slope of EF (m_EF) = (1 - 5) / (5 - 2) = -4 / 3.

There we have it, the slope of EF is -4/3. This looks promising, guys, because we've got a positive fraction and a negative fraction. The final segment we need to consider is FD, connecting F(5, 1) and D(-2, 2).

  • Slope of FD (m_FD) = (2 - 1) / (-2 - 5) = 1 / -7 = -1/7.

So, the slopes of our triangle's sides are m_DE = 3/4, m_EF = -4/3, and m_FD = -1/7. Now comes the crucial test: we need to see if the product of any two of these slopes equals -1. This is where we determine if any two sides are perpendicular, which is the defining characteristic of a right triangle.

Let's test the pairs:

  1. m_DE * m_EF: (3/4) * (-4/3) = -12 / 12 = -1.

Wow, would you look at that! We found a pair whose product is -1. This means that the side DE is perpendicular to the side EF. Because these two sides meet at a 90-degree angle at point E, the triangle DEF is indeed a right triangle.

Just for completeness, let's check the other pairs, though we already have our answer:

  1. m_DE * m_FD: (3/4) * (-1/7) = -3 / 28. This is not -1.
  2. m_EF * m_FD: (-4/3) * (-1/7) = 4 / 21. This is also not -1.

So, the only pair of slopes whose product is -1 is m_DE and m_EF. This confirms that the right angle is located at vertex E.

Why Other Conditions Don't Work

Now, let's quickly touch on why the other options provided in the original question aren't the correct condition for verifying a right triangle, even if they might be true for some right triangles. It's important to understand the definitive test.

A. Two of the sides of the triangle are equal.

This condition describes an isosceles triangle. While a right triangle can also be isosceles (this is called an isosceles right triangle, like a 45-45-90 triangle), not all right triangles are isosceles. For example, a triangle with side lengths 3, 4, and 5 is a right triangle (because 3^2 + 4^2 = 5^2, thanks to the Pythagorean theorem!), but all its sides have different lengths. Similarly, an isosceles triangle isn't necessarily a right triangle. Imagine a triangle with sides 5, 5, and 8. It's isosceles, but it doesn't have a 90-degree angle. So, having two equal sides is not the condition that universally verifies a triangle as a right triangle.

B. The product of the slopes of two of the sides is 1.

This is a common trap, guys! The condition for perpendicular lines (and thus a right angle in a triangle) is that the product of their slopes must be -1, not 1. If the product of two slopes is 1, it means the lines have a specific relationship, but it's not perpendicularity. For example, if one slope is 2, the other would need to be 1/2 for the product to be 1. These lines are neither parallel nor perpendicular. The negative sign in the '-1' product is crucial because it signifies the opposite and reciprocal relationship between the slopes of perpendicular lines (unless one line is horizontal and the other is vertical, which are special cases where one slope is 0 and the other is undefined).

The Correct Condition Recap

The definitive condition that verifies a triangle is a right triangle is that at least one pair of its sides must be perpendicular. In the coordinate plane, this translates to the product of the slopes of two of its sides must be -1. This is the mathematical cornerstone we used to solve our problem with triangle DEF. We calculated the slopes of DE, EF, and FD and found that the product of the slopes of DE (3/4) and EF (-4/3) was indeed -1. This confirmed that sides DE and EF are perpendicular, and therefore, triangle DEF is a right triangle with the right angle at vertex E. It’s a solid, reliable test that works every time, guys!

So, the next time you're faced with points on a coordinate plane and asked if they form a right triangle, you know exactly what to do: calculate the slopes, check their products, and look for that magic -1! Keep practicing, and you'll be a coordinate geometry whiz in no time. Stay curious and keep exploring the awesome world of math!