Math Problem Statement
Solution
Let's go over the details from the image of the triangle transformation problem.
Given:
- A right triangle is placed on a coordinate plane.
- The triangle is dilated by a factor of , centered at the origin.
- After dilation, it is translated 3 units to the right to form a new triangle .
Part A:
Find the area of the new triangle .
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First, calculate the area of the original triangle . The area of a triangle is given by: From the image, it looks like the base and height can be determined from the coordinates of points , , and .
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After dilation by , the new triangle’s dimensions are scaled down. Since the area scales by the square of the dilation factor, the area of will be:
Part B:
Find the length of the new side .
To determine this, first calculate the length of side using the distance formula: Then apply the dilation factor to find .
Let me go ahead and extract the necessary coordinates and solve these.### Solution:
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Part A:
- The area of the original triangle is square units.
- After dilation by a factor of , the area of the new triangle is:
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Part B:
- The length of side is approximately units.
- After dilation by , the length of the new side is:
Do you want further details on any of these parts or clarifications?
Related questions:
- How do transformations like translation affect the area of geometric shapes?
- What other transformations can be applied to a triangle besides dilation and translation?
- How is the distance formula derived, and how does it apply to different geometric shapes?
- Can you derive the relationship between dilation factors and area changes in general?
- How does the centroid of a triangle behave under dilation and translation?
Tip:
Always remember that when a shape is dilated by a factor , the area scales by , while the perimeter scales by .
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Math Problem Analysis
Mathematical Concepts
Dilation
Translation
Area of a Triangle
Coordinate Geometry
Formulas
Area of a Triangle = 1/2 * base * height
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Area scaling by dilation: New Area = (scale factor)^2 * Original Area
Theorems
Transformation Theorem
Dilation Theorem
Suitable Grade Level
Grades 9-10