Problem 65 - Algebra for College Students

Question

Solve each geometric figure problem. If a triangle has one right angle, then it is a right triangle. Use slope to determine whether the points \((-3,3),(-1,6)\) and \((0,0)\) are the vertices of a right triangle.

Step-by-Step Solution

Answer

The points do not form a right triangle.

Step 1: Find the slope of the line segment between points \((-3,3)\) and \((-1,6)\)

The slope of a line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, the points are \((-3,3)\) and \((-1,6)\). Substitute these coordinates into the formula to find the slope: \[ m_1 = \frac{6 - 3}{-1 + 3} = \frac{3}{2} \]

Step 2: Find the slope of the line segment between points \((-1,6)\) and \((0,0)\)

Using the same slope formula, calculate the slope between \((-1,6)\) and \((0,0)\): \[ m_2 = \frac{0 - 6}{0 + 1} = -6 \]

Step 3: Find the slope of the line segment between points \((0,0)\) and \((-3,3)\)

Again, using the slope formula for points \((0,0)\) and \((-3,3)\): \[ m_3 = \frac{3 - 0}{-3 - 0} = -1 \]

Step 4: Check for perpendicular slopes

For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, check the slopes calculated: \[ m_1 \times m_3 = \frac{3}{2} \times -1 = -\frac{3}{2} \]. This is not \(-1\), so these lines are not perpendicular. Check the other pairs: \[ m_2 \times m_3 = -6 \times -1 = 6 \]. This is also not \(-1\).

Step 5: Determine if the triangle is a right triangle

If none of the slopes are perpendicular, then the points \((-3,3),(-1,6)\) and \((0,0)\) do not form a right triangle.

Key Concepts

triangle properties

A triangle is a geometric figure with three sides and three angles. The sum of the interior angles of any triangle is always 180 degrees. There are different types of triangles classified based on their angles and sides.

• An equilateral triangle has all sides and angles equal.
• An isosceles triangle has two sides and two angles that are equal.
• A scalene triangle has all sides and angles different.
• A right triangle has one angle that is exactly 90 degrees.

The primary focus here is on the right triangle because it has a specific property where one angle is 90 degrees. This property creates a natural connection between the sides of the triangle, involving the Pythagorean theorem.
In this exercise, we use the concept of slopes of the sides to determine if the given set of points forms a right triangle.

slope formula

The slope of a line is a measure of how steep the line is. It's calculated by the 'rise' over the 'run,' or the vertical change divided by the horizontal change between two points on the line.
The formula for the slope (m) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's apply this formula to the points in our exercise:
For points \((-3,3)\) and \((-1,6)\):
\[ m_1 = \frac{6-3}{-1+3} = \frac{3}{2} \]
For points \((-1,6)\) and \((0,0)\):
\[ m_2 = \frac{0-6}{0+1} = -6 \]
For points \((0,0)\) and \((-3,3)\):
\[ m_3 = \frac{3-0}{-3-0} = -1 \]
These slopes help us understand the orientation of the sides of the triangle to determine if any of them are perpendicular.

geometric figures

Geometric figures include various shapes like triangles, squares, rectangles, and circles. Each shape has unique properties and characteristics.
The focus in this exercise is on the triangle, a fundamental geometric figure. A triangle's distinct features include:

• Three sides
• Three angles
• Three vertices (corner points where two sides meet)

A right triangle, which is of special interest in this problem, is characterized by one 90-degree angle. This specific kind of triangle is essential in many areas of math, including trigonometry and geometry.
The properties of geometric figures like triangles help in solving various mathematical problems, such as determining distances, angles, and areas.

perpendicular slopes

Two lines are perpendicular if they intersect at a right angle (90 degrees). For slopes, the lines are perpendicular if the product of their slopes is \(-1\).
Mathematically, if two lines have slopes \(m_1\) and \(m_2\), they are perpendicular if: \[ m_1 \times m_2 = -1 \]
Let's check the slopes from our exercise:

• \(m_1 = \frac{3}{2}\) and \(m_3 = -1\):
  \[ m_1 \times m_3 = \frac{3}{2} \times -1 = -\frac{3}{2} \]
  This is not \(-1\).
• \(m_2 = -6\) and \(m_3 = -1\):
  \[ m_2 \times m_3 = -6 \times -1 = 6 \]
  This is also not \(-1\).

As seen, none of the combinations of slopes result in \(-1\), meaning that the lines are not perpendicular. Therefore, these points do not form a right triangle.