Problem 15 - Prealgebra

Question

GEOMETRY Determine whether \(\triangle M N P\) with vertices \(M(3,-1), N(-3,2)\) and \(P(6,5)\) is isosceles. Explain your reasoning.

Step-by-Step Solution

Answer

Triangle MNP is isosceles because sides MN and MP are equal.

Step 1: Calculate Distance MN

Use the distance formula to calculate the length of segment \(MN\):\[MN = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For \(M(3, -1)\) and \(N(-3, 2)\):\[MN = \sqrt{(-3 - 3)^2 + (2 + 1)^2} = \sqrt{(-6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}\]

Step 2: Calculate Distance NP

Use the distance formula for segment \(NP\):\[NP = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}\]For \(N(-3, 2)\) and \(P(6, 5)\):\[NP = \sqrt{(6 + 3)^2 + (5 - 2)^2} = \sqrt{9^2 + 3^2} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10}\]

Step 3: Calculate Distance MP

Use the distance formula for segment \(MP\):\[MP = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}\]For \(M(3, -1)\) and \(P(6, 5)\):\[MP = \sqrt{(6 - 3)^2 + (5 + 1)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\]

Step 4: Determine if Triangle is Isosceles

In an isosceles triangle, at least two sides are of equal length. From steps 1 to 3, we see that \(MN = 3\sqrt{5}\), \(NP = 3\sqrt{10}\), and \(MP = 3\sqrt{5}\).The lengths \(MN\) and \(MP\) are equal, which confirms that \(\triangle MNP\) is isosceles.

Key Concepts

Distance Formula

When trying to determine side lengths in a geometric shape, the distance formula becomes very useful. Calculating the actual distance between two points in coordinate geometry can be done using the following formula:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula calculates the distance between any two points \((x_1, y_1)\text{ and }(x_2, y_2)\) in a two-dimensional space. The formula is derived from the Pythagorean theorem and gives us the straight-line distance, or Euclidean distance, between the points.

• First, find the difference in the x-coordinates and square it.
• Next, find the difference in the y-coordinates and square it.
• Add these squared differences together.
• Finally, take the square root of the sum to get the distance.

Vertices

Vertices are essentially the corners or points where two or more lines or edges meet in a shape. In the context of triangles, each vertex is formed at the meeting point of two sides.
For example, the triangle \(\triangle M N P\) has three vertices: \(M(3,-1)\text{, } N(-3,2)\text{, and } P(6,5)\). Each of these points defines where the lines of the triangle meet, forming its angles.

• The vertex \(M\) is formed by sides \(MN\) and \(MP\).
• The vertex \(N\) is where sides \(MN\) and \(NP\) intersect.
• The vertex \(P\) is the joining point of sides \(MP\) and \(NP\).

Vertices are particularly significant when determining properties of polygons, such as angle measurements, perimeter, and side lengths.

Isosceles Triangle Properties

An isosceles triangle is defined as a triangle that has at least two sides of equal length. This unique feature gives the triangle some special properties:

• Equal Angles: The angles opposite the equal sides are also equal. This means if two sides are the same, so are their opposite angles.
• Line of Symmetry: The isosceles triangle has one line of symmetry which runs from the vertex angle (where the equal sides meet) down to the midpoint of the base (the unequal side).
• Altitude, Median, and Angle Bisector: In an isosceles triangle, the altitude, median, and angle bisector to the base coincide, or are the same line.

These properties are helpful when solving problems related to isosceles triangles and can simplify many geometric calculations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to describe the position and properties of geometric figures using the coordinations on a graph.

In coordinate geometry, shapes like triangles, squares, and circles get defined using equational formats. The position of each vertex is identified by coordinates, such as \(M(3, -1)\text{, } N(-3, 2)\text{, } P(6, 5)\) for \(\triangle M N P\).

• Distance and Midpoint: Using the distance formula, one can find the length between points, and with the midpoint formula, find the average location between two points.
• Slope: The slope formula helps determine how steep a line is and the direction it goes unnoticed.
• Equation of a Line: With two points, we can find the equation of the line that passes through them, revealing important line information.

By using coordinate geometry, we can not only find important measurements but also apply algebraic techniques to solve geometric problems, offering a powerful toolkit for understanding and analyzing geometric configurations.