Calculating Lengths And Nature Of Triangle EFG

by Admin 47 views
Calculating Lengths and Nature of Triangle EFG

Hey guys! Let's dive into a fun math problem today where we'll be calculating the lengths of sides of a triangle and figuring out what kind of triangle it is. We're given three points in an orthonormal coordinate system: E(3, -2), F(-2, -3), and G(-3, 2). Our mission is to calculate the lengths of EF, FG, and EG, and then deduce the nature of the triangle EFG. So, grab your calculators and let's get started!

Calculating the Lengths of EF, FG, and EG

First things first, we need to find the distances between the points. Remember the distance formula? It's our trusty tool for calculating the distance between two points in a coordinate plane. The distance d between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)² + (y2 - y1)²)

Let's break it down step by step for each side of the triangle.

Calculating EF

To calculate the length of EF, we'll use the coordinates of points E(3, -2) and F(-2, -3). Plugging these values into the distance formula, we get:

EF = √((-2 - 3)² + (-3 - (-2))²) EF = √((-5)² + (-1)²) EF = √(25 + 1) EF = √26

So, the length of EF is √26 units. Easy peasy, right?

Calculating FG

Next up, let's find the length of FG using the coordinates of points F(-2, -3) and G(-3, 2). Here we go again with the distance formula:

FG = √((-3 - (-2))² + (2 - (-3))²) FG = √((-1)² + (5)²) FG = √(1 + 25) FG = √26

Guess what? The length of FG is also √26 units. Interesting!

Calculating EG

Now, for the final side, let's calculate the length of EG using the coordinates of points E(3, -2) and G(-3, 2). Time for the distance formula one more time:

EG = √((-3 - 3)² + (2 - (-2))²) EG = √((-6)² + (4)²) EG = √(36 + 16) EG = √52 EG = √(4 * 13) EG = 2√13

Alright, the length of EG is 2√13 units. We've got all the side lengths now!

Deducing the Nature of Triangle EFG

Now that we've calculated the lengths of all three sides (EF = √26, FG = √26, and EG = 2√13), we can figure out what type of triangle EFG is. There are a few clues here that we should pay attention to.

Isosceles Triangle Check

First, notice that EF and FG have the same length (√26). This means that triangle EFG has two sides of equal length. A triangle with two equal sides is called an isosceles triangle. So, we know EFG is at least an isosceles triangle.

Right Triangle Check

Next, let's see if EFG is a right triangle. To do this, we'll use the Pythagorean theorem. Remember the Pythagorean theorem? It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In simpler terms:

a² + b² = c²

Where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. The longest side is usually the hypotenuse, so let's check if EG² = EF² + FG²:

EG² = (2√13)² = 4 * 13 = 52 EF² = (√26)² = 26 FG² = (√26)² = 26

Now, let's see if the Pythagorean theorem holds:

52 = 26 + 26 52 = 52

It checks out! This means that triangle EFG is indeed a right triangle. The right angle is at vertex F, since EG (the longest side) is opposite to F.

Conclusion: Isosceles Right Triangle

So, we've determined that triangle EFG is both an isosceles triangle (because EF = FG) and a right triangle (because it satisfies the Pythagorean theorem). Therefore, triangle EFG is an isosceles right triangle. Awesome!

Diving Deeper into the Properties of Isosceles Right Triangles

Now that we've nailed down that EFG is an isosceles right triangle, let's chat a bit more about what makes these triangles special. Knowing these properties can sometimes give you shortcuts for solving other problems, or just help you appreciate the cool geometry we're dealing with!

Angles in an Isosceles Right Triangle

In any triangle, the angles add up to 180 degrees. In a right triangle, one of the angles is, by definition, 90 degrees. Now, in an isosceles right triangle, the other two angles (the ones that aren't the right angle) are equal. Let's figure out what those angles are. If we call one of these angles x, then we have:

90° + x + x = 180° 2x = 90° x = 45°

So, in an isosceles right triangle, the angles are 90 degrees, 45 degrees, and 45 degrees. These angles are super handy to recognize because they show up in lots of different math and science problems. It's like a mathematical fingerprint!

Side Ratios in an Isosceles Right Triangle

There’s another neat thing about isosceles right triangles – their sides have a special ratio. If the two equal sides each have a length a, then the hypotenuse has a length of a√2. We saw this in our original problem! We had EF = FG = √26, and EG = 2√13 = √(4 * 13) = √(2 * 2 * 13) = √(2 * 26) = √2 * √26.

This ratio is a direct result of the Pythagorean theorem and the fact that two sides are equal. Knowing this can save you time in calculations. If you know one side, you immediately know the others!

Applications in the Real World

Isosceles right triangles aren't just theoretical constructs; they show up in all sorts of real-world applications. Think about carpentry and construction – the 45-degree angle is a common one for making cuts and joining pieces. In navigation, they can help with determining directions and distances. Even in computer graphics and game development, these triangles are used for creating shapes and calculating angles.

Quick Recap and Tips

Before we wrap up, let's do a quick recap and share some tips for tackling these types of problems:

  1. Distance Formula is Your Friend: Remember the distance formula! It’s the key to calculating the lengths of the sides when you have coordinates.
  2. Pythagorean Theorem for the Win: Use the Pythagorean theorem to check if a triangle is a right triangle. It's a classic tool for a reason!
  3. Look for Equal Sides: If you spot two sides with the same length, you’re likely dealing with an isosceles triangle.
  4. Angles Add Up: Always remember that the angles in any triangle add up to 180 degrees.
  5. Practice Makes Perfect: The more problems you solve, the more comfortable you’ll get with these concepts. So, keep at it!

Conclusion: Mastering Geometric Calculations

So there you have it! We successfully calculated the lengths of the sides of triangle EFG and figured out that it's an isosceles right triangle. We also took a deeper dive into what makes these triangles tick, looking at their angles, side ratios, and real-world applications.

Remember, guys, math might seem tricky sometimes, but breaking it down step by step and using the right tools can make it a whole lot easier. Plus, understanding these geometric principles opens up a world of possibilities, from designing buildings to creating video games. Keep exploring, keep questioning, and most importantly, keep having fun with math! Until next time, happy calculating!