Math Problem Statement
A triangle has vertices D(-2,7), E(-4,2), and F(6,-2). a) Show algrebraically that this triangle is a right triangle. b) Find the midpoint of the hypotenuse. c) Show that this midpoint is equidistant from each of the vertices.
Solution
To solve this problem, we will approach it step by step, as follows:
Part (a): Show algebraically that the triangle is a right triangle.
To determine if the triangle is a right triangle, we need to check if any two sides are perpendicular by calculating the slopes of the sides. Two sides are perpendicular if the product of their slopes is .
The vertices of the triangle are:
Step 1: Calculate the slopes of the sides.
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Slope of DE:
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Slope of DF:
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Slope of EF:
Step 2: Check if the product of any two slopes is .
- Product of slopes of DE and EF: Since the product is , the sides and are perpendicular. Therefore, is a right triangle.
Part (b): Find the midpoint of the hypotenuse.
Since is a right triangle, the hypotenuse is the side opposite the right angle, which is .
The formula for the midpoint of a line segment between two points and is:
Here, the hypotenuse is , so the midpoint is:
Part (c): Show that the midpoint is equidistant from each of the vertices.
To prove that the midpoint is equidistant from each of the vertices , , and , we need to calculate the distances between the midpoint and each vertex, and show that they are equal.
The distance formula between two points and is:
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Distance from the midpoint to :
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Distance from the midpoint to :
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Distance from the midpoint to :
Since the distances are equal, the midpoint is equidistant from all three vertices.
Final Answer:
- is a right triangle because the slopes of and are perpendicular.
- The midpoint of the hypotenuse is .
- The midpoint is equidistant from each of the vertices, with a distance of units from each.
Would you like more details on any part of this problem?
Here are five related questions to explore:
- How do you calculate the distance between two points in 3D space?
- What is the formula for the area of a right triangle given its vertices?
- How can you find the equation of the line through two points?
- What are the properties of midpoints in different types of triangles?
- How can the Pythagorean theorem be used to verify right triangles?
Tip: Always double-check your slope calculations when verifying perpendicularity, as this is crucial to identifying right triangles.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Distance Formula
Slope
Midpoint Formula
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Midpoint formula: ((x1 + x2) / 2, (y1 + y2) / 2)
Distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Pythagorean Theorem
Right Triangle Theorem
Suitable Grade Level
Grades 9-10