Math Problem Statement

Right triangle TUV, shown in the coordinate plane below, is dilated by a factor of 1/2, centered at the origin, and then translated 3 units to the right to form a new triangle T'U'V'. What is the area of the new triangle T'U'V'? What is the length of the new side U'V'?

Solution

Let's go over the details from the image of the triangle transformation problem.

Given:

  1. A right triangle TUVTUV is placed on a coordinate plane.
  2. The triangle is dilated by a factor of 12\frac{1}{2}, centered at the origin.
  3. After dilation, it is translated 3 units to the right to form a new triangle TUVT'U'V'.

Part A:

Find the area of the new triangle TUVT'U'V'.

  1. First, calculate the area of the original triangle TUVTUV. The area of a triangle is given by: Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} From the image, it looks like the base and height can be determined from the coordinates of points TT, UU, and VV.

  2. After dilation by 12\frac{1}{2}, the new triangle’s dimensions are scaled down. Since the area scales by the square of the dilation factor, the area of TUVT'U'V' will be: New Area=(12)2×Original Area\text{New Area} = \left(\frac{1}{2}\right)^2 \times \text{Original Area}

Part B:

Find the length of the new side UVU'V'.

To determine this, first calculate the length of side UVUV using the distance formula: Distance=(x2x1)2+(y2y1)2\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Then apply the dilation factor to find UVU'V'.


Let me go ahead and extract the necessary coordinates and solve these.### Solution:

  1. Part A:

    • The area of the original triangle TUVTUV is 1616 square units.
    • After dilation by a factor of 12\frac{1}{2}, the area of the new triangle TUVT'U'V' is: 4 square units.\boxed{4} \text{ square units}.
  2. Part B:

    • The length of side UVUV is approximately 5.665.66 units.
    • After dilation by 12\frac{1}{2}, the length of the new side UVU'V' is: 2.83 units.\boxed{2.83} \text{ units}.

Do you want further details on any of these parts or clarifications?


Related questions:

  1. How do transformations like translation affect the area of geometric shapes?
  2. What other transformations can be applied to a triangle besides dilation and translation?
  3. How is the distance formula derived, and how does it apply to different geometric shapes?
  4. Can you derive the relationship between dilation factors and area changes in general?
  5. How does the centroid of a triangle behave under dilation and translation?

Tip:

Always remember that when a shape is dilated by a factor kk, the area scales by k2k^2, while the perimeter scales by kk.

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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