A dilation enlarges or reduces a figure with respect to a fixed center point, producing an image that is similar to — but not congruent to — the pre-image. This page defines the scale factor as the ratio of image to pre-image lengths, works through dilations centered at the origin and at any other point, and distinguishes enlargements from reductions. Watch the direction of the ratio: image divided by pre-image, and the two directions are reciprocals of each other.
What you are looking at: a coordinate grid with a triangle ABC and its image A′B′C′ under a dilation whose center is the origin O. A faint ray from the origin runs through each vertex to its image, and the segment from the origin to the selected image vertex is highlighted.
What to do: press a scale factor button to change k, and press a vertex button to select that vertex. The box under the grid shows the coordinates of the selected point and its image, the two distances from the center, and their ratio.
What to notice: every vertex moves along its own ray from the center, and every distance from the center is multiplied by the same k. When k is greater than 1 the image is an enlargement; when k is between 0 and 1 the image is a reduction; when k = 1 the image coincides with the pre-image and is congruent to it.
The scale factors above are positive, and almost every dilation in this lesson uses a positive scale factor. The two buttons below are for observation only — there is nothing to score here. A dilation with k = −2 sends each point to the opposite ray at twice the distance from the center: the image of P is on the other side of the center from P. That is exactly the result of a composition — a dilation with scale factor 2 followed by a rotation of 180° about the center. Press a button, compare the image with the one for k = 2, and notice that the two images have the same size and shape and differ only by that half-turn about the center.
What you are looking at: a triangle ABD, a center of dilation O that is not the origin, and a scale factor. The method from the notes has three steps: translate so the center lands at the origin, multiply every coordinate by k, then translate back.
What to do: answer the three steps in order. Each step asks about one point of the triangle, and the diagram updates after each step so you can watch the triangle move to the origin, change size, and move back. A first wrong answer earns a hint and one retry; after a second miss the correct choice is shown and the step is complete. When you finish an item, press Next Item for a new center, a new triangle, and a new scale factor.
What to notice: the middle step is the same origin rule (x, y) → (kx, ky) from the activity above. The two translations exist only to move the center to the origin and back, and they undo each other.
What you are looking at: a pre-image triangle in blue, a dashed target outline, and a marked center of dilation. The target is the image of the pre-image under one dilation about that center.
What to do: first choose the scale factor k that sends the pre-image exactly onto the target — one of the offered values is the reciprocal of the correct one, so check the direction of the ratio — then say whether the dilation is an enlargement or a reduction. When both parts are settled, the triangle scales onto the target and lands exactly on it. Press New Challenge for a new item. Later items use a center that is not the origin.
What to notice: the correct k can be found from any pair of corresponding vertices: divide the distance from the center to the image vertex by the distance from the center to the pre-image vertex.
Six questions, one at a time. A first wrong answer earns a hint and one retry; the full explanation appears when a question is answered correctly or after a second miss. Your score counts first-try answers only.
Lesson 36 · Dilations · Mr. J Geometry. All activities on this page run entirely in your browser.