Question

Find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (5,3,4),(7,1,3),(3,5,3)

Short answer

The lengths of the sides are 3, 3, and \( 4 \sqrt{2} \). It is an isosceles but not a right triangle.

Step-by-step solution

Calculate Length of Side AB

Using the distance formula in 3D Space: \( \sqrt{(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2} \), calculate the distance between A(5,3,4) and B(7,1,3). This would result in AB = \( \sqrt{(7-5)^2 + (1-3)^2 + (3-4)^2} \) = \( \sqrt{4+4+1} \) = \( \sqrt{9} \) = 3.

Calculate Length of Side BC

Now, calculate the distance between B(7,1,3) and C(3,5,3) in the same way. BC = \( \sqrt{(3-7)^2 + (5-1)^2 + (3-3)^2} \) = \( \sqrt{16+16+0} \) = \( \sqrt{32} \) = \( 4 \sqrt{2} \)

Calculate Length of Side AC

Calculate the distance between A(5,3,4) and C(3,5,3). AC = \( \sqrt{(3-5)^2 + (5-3)^2 + (3-4)^2} \) = \( \sqrt{4+4+1} \) = \( \sqrt{9} \) = 3.

Determine Type of Triangle

Now that the lengths of the sides are found, evaluate the type of triangle. Here, sides AB and AC are equal, implying that ABC is an isosceles triangle. It's also necessary to use the Pythagorean theorem to check whether the triangle is right angled. If \( AB^2 + AC^2 = BC^2 \), then it's a right triangle. For triangle ABC, \( 3^2 + 3^2 = (4 \sqrt{2})^2 \). Solving this, we get \( 18 \neq 32 \), so it's not a right triangle.

Key concepts

Distance Formula in 3D Space

Determining the precise distance between two points in 3D space is essential for various applications, including geometry. The distance formula \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \) is a direct extension of the Pythagorean theorem into three dimensions. It incorporates the changes along each axis between two points, A(5,3,4) and B(7,1,3) as an example.

When applying this formula, each term under the square root sign represents the square of the distance between the coordinates of the two points along the respective axis. Visualizing this in three-dimensional space can be thought of as finding the hypotenuse of a right-angled triangle, but in 3D, where there are three mutually perpendicular sides to consider instead of two.

Isosceles Triangle Characteristics

The characteristics of an isosceles triangle are intriguing and crucial to understanding. These triangles have at least two sides of equal length, which we refer to as the 'legs' of the triangle. The angles opposite these legs are also equal. In our example, sides AB and AC have the same length.

An isosceles triangle can be visualized as having a line of symmetry through the vertex angle and down to the midpoint of the base, the side that is not equal in length. This symmetry results in the equal angles and the remarkable fact that if one were to draw an altitude from the vertex angle to the base, it would not only bisect the base but also the vertex angle. Understanding these properties helps in solving geometry problems and identifying the types of triangles.

Pythagorean Theorem

The Pythagorean theorem, a fundamental principle in geometry, relates the lengths of the sides of a right triangle. The theorem states that for a right triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This relationship is expressed algebraically as \( a^2 + b^2 = c^2 \).

Applying this theorem helps determine whether a triangle is a right triangle. In the exercise, by calculating the squares of the lengths of the sides and comparing them, \( 3^2 + 3^2 ≠ (4 \sqrt{2})^2 \) shows that the triangle in question is not a right triangle. It's a powerful tool in various mathematical and real-world applications, such as construction, navigation, and computer graphics.