Choose The Correct Scale Factor

Effects of Dilations on Length, Surface area, and Angles

Alignments to Content Standards: 8.Thou.A.iii

Task

Consider triangle $ABC$.Â

1. Describe a dilation of $ABC$ with:
   1. Middle $A$ and scale factor 2.
   2. Heart $B$ and calibration factor 3.Â
   3. Middle $C$ and scale factor $\frac12$.
2. For each dilation, answer the following questions:

   Â

   1. Past what factor practice the base and height of the triangle change? Explain.
   2. By what gene does the surface area of the triangle change? Explain.
   3. How do the angles of the scaled triangle compare to the original? Explicate.

IM Commentary

The purpose of this task is for students to study the impact of dilations on different measurements: segment lengths, area, and bending mensurate. When a triangle is dilated by calibration factor $southward \gt 0$, the base and height change past the scale cistron $southward$ while the expanse changes by a factor of $south^2$: as seen in the examples presented here, this is truthful regardless of the center of dilation. While they calibration distances between points, dilations do not change angles. While $10$ and $y$ coordinates have not been given to the vertices of the triangle, the coordinate grid serves the same purpose for the given centers of dilation.

Transformations affect all points in the plane, not just the particular figures we choose to analyze when working with transformations. All lengths of line segments in the plane are scaled by the same factor when we employ a dilation. Students can use a variety of tools with this task including colored pencils, highlighters, graph paper, rulers, protractors, and/or transparencies. Job 1681 would exist a good follow up to this job, particularly if students take access to dynamic geometry software, where they tin see that this is true for arbitrary triangles.

Solution

1. The iii dilations are shown below along with explanations for the pictures:

   1. Â

      The dilation with center $A$ and scale factor two doubles the length of segments $\overline{AB}$ and $\overline{Air-conditioning}$. We can see this explicitly for $\overline{AC}$. For $\overline{AB}$, this segment goes over 6 units and up four so its image goes over 12 units and up eight units.Â

   2. Â

      The dilation with center $B$ and scale factor 3 increases the length of $\overline{AB}$ and $\overline{Ac}$ past a factor of 3. The point $B$ does not motility when nosotros use the dilation only $A$ and $C$ are mapped to points 3 times as far from $B$ on the same line.

   3. Â

   

   The scale factor of $\frac{ane}{2}$ makes a smaller triangle. The center of this dilation (as well called a contraction in this case) is $C$ and the vertices $A$ and $B$ are mapped to points half the altitude from $A$ on the aforementioned line segments.Â

2. 1. When the scale factor of two is applied with center $A$ the length of the base doubles from 6 units to 12 units. This is also truthful for the height which was four units for $\triangle ABC$ only is 8 units for the scaled triangle. Similarly, when the scale factor of 3 is practical with middle $B$ , the length of the base and the pinnacle increase by a scale factor of three and for the calibration factor of $\frac{1}{two}$ with heart $C$, the base and elevation of $\triangle ABC$ are likewise scaled by $\frac{1}{2}$.

   2. The surface area of a triangle is the base of operations times the height. When a scale cistron of ii with center $A$ is applied to $\triangle ABC$, the base and superlative each double so the expanse increases by a factor of 4: the surface area of $\triangle ABC$ is 12 square units while the expanse of the scaled version is 48 square units. Similarly, if a calibration factor of 3 with center $B$ is applied then the base and height increase by a factor of 3 and the area increased past a gene of 9. Finally, if a scale factor of 1/2 with center $C$ is applied to $\triangle ABC$, the base and acme are cutting in half and then the area is multiplied by i/4.

   3. The angle measures do not change when the triangle is scaled. For the offset scaling, we tin meet that bending $A$ is common to $\triangle ABC$ and its scaling with middle $A$ and scaling factor 2. Angle $B$ is congruent to its scaled image equally we tin see by translating $\triangle ABC$ eight units to the right and 4 units up. Finally, angle $C$ is congruent to its scaled epitome equally nosotros verify past translating $\triangle ABC$ 8 units to the right.Â

Furnishings of Dilations on Length, Surface area, and Angles

Consider triangle $ABC$.Â

1. Draw a dilation of $ABC$ with:
   1. Centre $A$ and scale gene two.
   2. Center $B$ and scale cistron 3.Â
   3. Center $C$ and scale gene $\frac12$.
2. For each dilation, answer the following questions:

   Â

   1. By what gene do the base and tiptop of the triangle change? Explicate.
   2. By what gene does the expanse of the triangle change? Explicate.
   3. How do the angles of the scaled triangle compare to the original? Explain.

Choose The Correct Scale Factor,

Source: https://tasks.illustrativemathematics.org/content-standards/tasks/1682

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