Grade 10 · Core Mathematics · Lesson 2 of 11

Finding the Central Point of Dilation and Scale Factors

A Grade 10 Core Mathematics CBE lesson from the sub-strand Similarity and Enlargement. In it, learners locate the fixed central point by constructing rays through matching vertices of object and image figures.

Strand Measurements and GeometrySub-strand Similarity and EnlargementLesson code 10CM-21-L240 minutes · 2 slides

KICD learning outcome (a)

Determine the centre of enlargement and the linear scale factor for similar figures.

By the end of this lesson, learners should be able to

Locate the fixed central point by constructing rays through matching vertices of object and image figures.
Calculate positive and negative scale factors for given scale transformations on a Cartesian grid.

Key vocabulary

Centre of enlargement: a fixed central point from which all figure coordinates expand or contract.; Drawing lines through matching vertices reveals this central point at their intersection.
Negative scale factor: a ratio where the image is inverted and placed across the central point.; An inverted image positioned opposite the focal point yields a negative multiplier value.

Lesson preview

Slide 1 · Locating Central Expansion Point and Negative Scale FactorsModel · 18 min

Figure 1: A coordinate graph displaying object triangle ABC and its inverted image A'B'C', with dashed ray lines intersecting at the origin.

To locate the central expansion point geometrically, construct straight lines joining each object point to its matching image point. The location where all these ray lines cross is the focal central point. When the object and image lie on opposite sides of this intersection location, the resulting image is inverted and the associated scale factor is negative.

Joining matching object and image vertices creates straight rays that meet at the central focal point.
The scale factor equals the ratio of any image side length to its matching object side length.
An inverted image located on the opposite side of the central point indicates a negative multiplier.
When the image is identical in size but inverted across the focal point, the scale factor is -1.

Worked example. Step 1: Draw straight rays joining matching vertices: AA' through (2, 3) and (-2, -3), BB' through (5, 3) and (-5, -3), and CC' through (2, 7) and (-2, -7). Step 2: Find the point of intersection of these rays. All three lines cross at the origin O(0, 0). Step 3: Measure matching side lengths: AB = 5 - 2 = 3 units, and A'B' = |-5 - (-2)| = 3 units. Step 4: Calculate the scale factor ratio: A'B' / AB = 3 / 3 = 1. Since the image is inverted across O(0, 0), the scale factor is -1. Answer: Central point of expansion is (0, 0) and linear scale factor is -1..

Check question: Triangle $P Q R$ with $P (3, 4)$ is enlarged to $P^{'} (- 6, - 8)$ through the origin $O (0, 0)$ . If $Q R = 4 cm$ and $Q^{'} R^{'} = 8 cm$ with an inverted orientation, what is the scale factor of this enlargement?

-2
2
-0.5
0.5

Show answer

-2. The magnitude of the scale factor is the ratio of the image length to the object length: $8 / 4 = 2$ . Because the image is inverted across the origin (which can also be seen since the coordinates of $P^{'}$ are exactly $- 2$ times the coordinates of $P$ ), the scale factor must be negative. Thus, the scale factor is $- 2$ .

Slide 2 · Determining Enlargement Parameters in PracticePractice · 22 min

Figure 2: A coordinate grid illustrating the enlargement of triangle MKL to M'K'L', with solid rays intersecting at the centre of dilation.

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Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Finding the Central Point of Dilation and Scale Factors

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Similarity and Enlargement