| Front | Back | | --- | --- | | What does it mean for two figures to be **similar** in geometry? | Two figures are **similar** when: 1. Their **corresponding angles are equal**. 2. All **corresponding side lengths are in the same ratio**, so each side is multiplied by the same scale factor. | | How do we write that figure A is similar to figure B? | We write **$A ext{ } ilde{} ext{ } B$** (read as "A is similar to B"). | | What is a **scaled copy** of a shape? | A **scaled copy** is a figure that has the same shape as the original, but a different size (and possibly a different position or orientation), with all side lengths multiplied by the same scale factor. | | What is a **dilation** in geometry? | A **dilation** (enlargement or reduction) is a transformation that changes the size of a figure but keeps its shape. Angles stay the same and side lengths are scaled by a constant factor. | | What is the **scale factor** $r$ in a dilation? | The **scale factor** $r$ is the ratio of a side length in the image to the corresponding side length in the original figure. It tells how many times longer or shorter each length becomes. | | How does the size of a figure change when $|r|>1$ in a dilation? | When $|r|>1$, the dilation is an **enlargement**: every length in the image is greater than in the original by a factor of $|r|$. | | How does the size of a figure change when $0<|r|<1$ in a dilation? | When $0<|r|<1$, the dilation is a **reduction**: every length in the image is smaller than in the original by a factor of $|r|$. | | If a figure is dilated by scale factor $r$, how do its **lengths** change? | Every **length** (side, diagonal, perimeter) is multiplied by $|r|$. | | If a figure is dilated by scale factor $r$, how do its **area** and **volume** change? | - Every **area** is multiplied by $|r|^2$. - Every **volume** (in 3D) is multiplied by $|r|^3$. | | What is the **center of dilation**? | The **center of dilation** is the fixed point $C$ such that, in a dilation, each point $P$ in the original is mapped to a point $P'$ on the line $CP$, and $CP'$ is $|r|$ times $CP$. | | How are points related to the center of dilation in a dilation with scale factor $r$ and center $C$? | For each original point $P$: 1. $P'$ lies on the line joining $C$ and $P$. 2. The distance $CP'$ equals $|r| imes CP$. 3. If $r>0$, $P'$ is on the **same side** of $C$ as $P$. 4. If $r<0$, $P'$ is on the **opposite side** of $C$ from $P$. | | What extra effect occurs when a dilation has a **negative scale factor**? | A dilation with a **negative scale factor** includes a **half-turn (180° rotation)** about the center of dilation, as well as the size change by factor $|r|$. | | What is the coordinate rule for a dilation from the **origin** with scale factor $r$? | If the center of dilation is the origin $(0,0)$ and the scale factor is $r$, then: $$(x,y) ightarrow (rx, ry)$$ | | What is the coordinate rule for a dilation with center $C(a,b)$ and scale factor $r$? | If the center of dilation is $C(a,b)$ and the scale factor is $r$, then: $$(x,y) ightarrow (a + r(x-a), ext{ } b + r(y-b))$$ This scales the point relative to the chosen center $(a,b)$. | | How does a dilation show that the image is **similar** to the original figure? | In a dilation: 1. All corresponding **angles stay equal**. 2. All corresponding **side lengths are multiplied by $r$**, so $$\frac{\text{image side}}{\text{original side}} = |r|$$ for every pair of corresponding sides. Therefore, the image is **similar** to the original. | | What are the three main similarity tests for **triangles**? | 1. **AA** (angle-angle): two corresponding angles equal. 2. **SSS proportional**: three pairs of corresponding sides are in the same ratio. 3. **SAS proportional**: two pairs of corresponding sides are in the same ratio and the included angle is equal. | | How does a **positive** scale factor $r>0$ affect the orientation and position of the image? | If $r>0$: 1. The image lies on the **same side** of the center as the original. 2. Orientation (clockwise vs anticlockwise) is **preserved**. 3. If $|r|>1$ it is an enlargement, if $0<|r|<1$ it is a reduction. | | How does a **negative** scale factor $r<0$ affect the orientation and position of the image? | If $r<0$: 1. The image lies on the **opposite side** of the center from the original. 2. The figure is still similar and scaled by $|r|$. 3. There is a **180° rotation** about the center (orientation is reversed). | | If two similar figures have side-length ratio $\dfrac{3}{5}$, what is the ratio of their **areas**, and is this an enlargement or reduction? | - The area ratio is $\left(\dfrac{3}{5}\right)^2 = \dfrac{9}{25}$. - Because $\dfrac{3}{5}<1$, this represents a **reduction** from the larger figure to the smaller one. | | A quadrilateral is dilated with scale factor $r=-\dfrac12$ and center at $C$. How do the **size** and **position** of the image compare to the original? | - **Size**: Every length is multiplied by $\left| -\dfrac12 \right| = \dfrac12$, so the image is half as large (a reduction by factor $\dfrac12$). Its area is multiplied by $\left(\dfrac12\right)^2 = \dfrac14$. - **Position**: Each image point lies on the line through $C$ and the original point, but on the **opposite side** of $C$, and the figure has been rotated by 180° about $C$. |

Advanced Number and Bounds3 subtopics

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Function Transformations4 subtopics

Advanced Probability1 subtopic

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Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

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MYP Extended Mathematics Similar Shapes and Dilation Flashcards