Triangle EFG is dilated with a scale factor of to create . The measure of is 36ï‚°. What is ?

Effects of Dilations on Length, Area, and Angles

Alignments to Content Standards: eight.Chiliad.A.three

Task

Consider triangle $ABC$.Â

1. Draw a dilation of $ABC$ with:
   1. Center $A$ and calibration factor 2.
   2. Heart $B$ and calibration factor 3.Â
   3. Heart $C$ and calibration cistron $\frac12$.
2. For each dilation, reply the following questions:

   Â

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

IM Commentary

The purpose of this job is for students to report the touch on of dilations on dissimilar measurements: segment lengths, area, and angle measure. When a triangle is dilated by scale factor $south \gt 0$, the base and height change by the scale factor $s$ while the expanse changes by a factor of $due south^two$: as seen in the examples presented here, this is truthful regardless of the center of dilation. While they scale distances between points, dilations exercise not change angles. While $10$ and $y$ coordinates have non been given to the vertices of the triangle, the coordinate grid serves the aforementioned purpose for the given centers of dilation.

Transformations impact all points in the plane, not simply the item figures we choose to analyze when working with transformations. All lengths of line segments in the plane are scaled by the aforementioned cistron when we apply a dilation. Students tin can utilise a variety of tools with this task including colored pencils, highlighters, graph paper, rulers, protractors, and/or transparencies. Task 1681 would be a good follow upward to this chore, specially if students have access to dynamic geometry software, where they can see that this is truthful for arbitrary triangles.

Solution

1. The three dilations are shown beneath along with explanations for the pictures:

   1. Â

      The dilation with center $A$ and calibration factor 2 doubles the length of segments $\overline{AB}$ and $\overline{AC}$. We can see this explicitly for $\overline{Ac}$. For $\overline{AB}$, this segment goes over 6 units and up 4 so its image goes over 12 units and up eight units.Â

   2. Â

      The dilation with center $B$ and scale cistron 3 increases the length of $\overline{AB}$ and $\overline{Ac}$ by a factor of 3. The betoken $B$ does non move when nosotros apply the dilation but $A$ and $C$ are mapped to points 3 times as far from $B$ on the same line.

   3. Â

   

   The calibration cistron of $\frac{ane}{2}$ makes a smaller triangle. The heart of this dilation (also called a wrinkle in this case) is $C$ and the vertices $A$ and $B$ are mapped to points half the altitude from $A$ on the same line segments.Â

2. 1. When the calibration cistron of 2 is applied with eye $A$ the length of the base doubles from 6 units to 12 units. This is likewise true for the height which was 4 units for $\triangle ABC$ just is 8 units for the scaled triangle. Similarly, when the calibration factor of 3 is applied with center $B$ , the length of the base and the acme increase by a scale factor of 3 and for the calibration cistron of $\frac{1}{2}$ with center $C$, the base of operations and height of $\triangle ABC$ are likewise scaled by $\frac{1}{2}$.

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

   3. The angle measures practise not change when the triangle is scaled. For the showtime scaling, we can see that angle $A$ is common to $\triangle ABC$ and its scaling with center $A$ and scaling cistron 2. Angle $B$ is congruent to its scaled image as nosotros tin can see by translating $\triangle ABC$ 8 units to the right and four units up. Finally, angle $C$ is congruent to its scaled image as we verify by translating $\triangle ABC$ 8 units to the right.Â

Effects of Dilations on Length, Expanse, and Angles

Consider triangle $ABC$.Â

1. Draw a dilation of $ABC$ with:
   1. Center $A$ and scale factor ii.
   2. Center $B$ and scale factor iii.Â
   3. Center $C$ and scale factor $\frac12$.
2. For each dilation, reply the following questions:

   Â

   1. By what factor practise the base and height of the triangle alter? Explain.
   2. By what cistron does the area of the triangle change? Explicate.
   3. How practice the angles of the scaled triangle compare to the original? Explain.

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

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