Question

Determine whether \(\triangle F G H\) with vertices \(F(-2,1), G(1,6)\), and \(H(4,1)\) is isosceles.

Short answer

Yes, \( \triangle FGH \) is isosceles as \( FG = GH = \sqrt{34} \).

Step-by-step solution

Find the distance between F and G

Use the distance formula to find the length of \( FG \). The distance formula is \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Substitute \( F(-2,1) \) and \( G(1,6) \) into the formula: \( FG = \sqrt{(1 - (-2))^2 + (6 - 1)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \).

Find the distance between G and H

Use the distance formula again to find the length of \( GH \). Substitute \( G(1,6) \) and \( H(4,1) \) into the formula: \( GH = \sqrt{(4 - 1)^2 + (1 - 6)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \).

Find the distance between H and F

Use the distance formula to find the length of \( FH \). Substitute \( H(4,1) \) and \( F(-2,1) \) into the formula: \( FH = \sqrt{(4 - (-2))^2 + (1 - 1)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 \).

Compare the distances

Compare the lengths of \( FG, GH, \) and \( FH \). We found that \( FG = GH = \sqrt{34} \) and \( FH = 6 \). Therefore, \( FG \) and \( GH \) are equal, indicating that \( \triangle FGH \) is isosceles.

Key concepts

Distance Formula

The distance formula is a handy tool in coordinate geometry that helps us calculate the distance between two points in a plane. It's particularly useful when working with triangles, as it lets us determine the lengths of the sides when their vertices are given. The formula is expressed as:

• \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)

Here,

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

To apply it, simply subtract the coordinates of the first point from the second, square the differences, add those squared values together, and then take the square root.
For example, to find the length of side \( FG \) in \( \triangle FGH \) with \( F(-2,1) \) and \( G(1,6) \), plug the coordinates into the formula:
\( FG = \sqrt{(1 - (-2))^2 + (6 - 1)^2} = \sqrt{3^2 + 5^2} = \sqrt{34} \).
This consistent method is vital for confirming triangle properties and can be used for any pair of points you encounter.

Triangle Properties

Triangles come in various types based on their side lengths and angles. An isosceles triangle is one of the most studied triangles, known for having at least two equal sides.
This property is important in identifying and solving problems involving triangles. In an isosceles triangle, because two sides are of equal length, two angles opposite those sides are also equal.
In our example, by proving \( FG = GH \) being equal confirms that \( \triangle FGH \) is isosceles.
Thus, it maintains the symmetry in its structure and calculations.
Understanding these properties allows us to work on more complex problems involving bisecting angles and calculating areas without direct measurements.

• The base angles are equal in an isosceles triangle.
• An isosceles triangle can be further classified as acute, right, or obtuse based on its angles.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, extends traditional geometry into a coordinate plane. It seamlessly integrates algebra into geometry by using coordinates to represent geometric shapes.
This approach simplifies the process of solving and analyzing geometric problems by translating them into algebraic equations.

• It allows calculation of distances between points using the distance formula.
• It is instrumental in finding midpoints through the midpoint formula: \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
• Helps in verifying the type of a triangle based on its side lengths.

By placing points \( F(-2,1) \), \( G(1,6) \), and \( H(4,1) \) in a coordinate plane, we can easily compute and validate the properties of \( \triangle FGH \).
Coordinate geometry doesn't just make calculations easier; it also enhances the visualization of geometric concepts. It creates a bridge between visual shapes and numerical data, providing a comprehensive understanding of geometric relationships.