A triangle with an area of $133$ units $^{2}$ is the image of a triangle that was dilated by a scale factor of $\frac{3}{4}$ . Find the area of the preimage, the original triangle, before its dilation. Round your answer to the nearest tenth, if necessary. $\newline$ Answer: units $^{2}$

View step-by-step help

Home

Math Problems

Geometry

Dilations and parallel lines

Full solution

Q. A triangle with an area of

133

units

^{2}

is the image of a triangle that was dilated by a scale factor of

\frac{3}{4}

. Find the area of the preimage, the original triangle, before its dilation. Round your answer to the nearest tenth, if necessary.

\newline

Answer: units

^{2}

Understand Relationship: To find the area of the original triangle before dilation, we need to understand the relationship between the area of the dilated triangle and the area of the original triangle. The area of the dilated triangle is equal to the square of the scale factor times the area of the original triangle. $\newline$ Mathematically, this is represented as: $\newline$ Area of dilated triangle $= (\text{Scale factor})^2 \times \text{Area of original triangle}$
Set Up Equation: Given that the area of the dilated triangle is $133$ square units and the scale factor is $\frac{3}{4}$ , we can set up the equation: $\newline$ $133 = \left(\frac{3}{4}\right)^2 \times \text{Area of original triangle}$
Calculate Scale Factor: To find the area of the original triangle, we need to divide the area of the dilated triangle by the square of the scale factor. $\newline$ Area of original triangle = $\frac{133}{(\frac{3}{4})^2}$
Divide by Reciprocal: Calculating the square of the scale factor $(\frac{3}{4})^2$ gives us: $\newline$ $(\frac{3}{4})^2 = (\frac{3}{4}) \times (\frac{3}{4}) = \frac{9}{16}$
Perform Multiplication: Now we divide the area of the dilated triangle by the square of the scale factor to find the area of the original triangle: $\newline$ Area of original triangle = $\frac{133}{(\frac{9}{16})}$
Calculate Division: To divide by a fraction, we multiply by its reciprocal. So, we multiply $133$ by the reciprocal of $\frac{9}{16}$ , which is $\frac{16}{9}$ : $\newline$ Area of original triangle = $133 \times \left(\frac{16}{9}\right)$
Calculate Division: To divide by a fraction, we multiply by its reciprocal. So, we multiply $133$ by the reciprocal of $\frac{9}{16}$ , which is $\frac{16}{9}$ : $\newline$ Area of original triangle = $133 \times \left(\frac{16}{9}\right)$ Performing the multiplication gives us: $\newline$ Area of original triangle = $133 \times 16 / 9$ $\newline$ Area of original triangle = $\frac{2128}{9}$
Calculate Division: To divide by a fraction, we multiply by its reciprocal. So, we multiply $133$ by the reciprocal of $\frac{9}{16}$ , which is $\frac{16}{9}$ :
Area of original triangle = $133 \times \left(\frac{16}{9}\right)$ Performing the multiplication gives us:
Area of original triangle = $133 \times 16 / 9$
Area of original triangle = $2128 / 9$ Finally, we calculate the division to find the area of the original triangle:
Area of original triangle = $2128 / 9 \approx 236.4$