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Example Question - scale factor

Here are examples of questions we've helped users solve.

Determining Scale Factor in Similar Triangles

The given problem involves similar figures, specifically two triangles for which you must find the scale factor. To determine the scale factor from one triangle to another, you can divide a side length of one triangle by the corresponding side length of the other triangle. In the image, Triangle CDE with sides CD = 10, DE = 12, and CE = 16, and Triangle VUT with sides UV = 9.6 and UT = 12 are shown. Since the triangles are similar, their corresponding sides are proportional. We will take the side lengths of Triangle VUT and divide them by the corresponding sides of Triangle CDE to find the scale factor: VU (UV) / CD = 9.6 / 10 = 0.96 UT / CE = 12 / 16 = 0.75 As you can see, we got two different results for the scale factor which should not happen in similar figures. Since we only need two sides to determine the scale factor, let's assume the correct proportion for similar triangles using sides VU to CD and UT to DE, which are the more likely corresponding sides because they match the orientations of the triangles: VU (UV) / CD = 9.6 / 10 = 0.96 UT / DE = 12 / 12 = 1 In this case, using sides UT and DE, we receive a scale factor of 1, which is contradictory to the notion of a scale factor that's supposed to reduce or enlarge the figures. Given that similar figures should have the same scale factor for all corresponding sides, there seems to be an inconsistency in the given side lengths. Based on these calculations and the apparent inconsistency, there might be an error in the values provided or in the assumption of which sides correspond. If we disregard the irregularity and presume the intention was for the sides to be proportional, then the scale factor based on side UV to CD is 0.96. However, it's important to consult with an instructor or the source of the problem to clarify the correct corresponding sides and values to determine the intended scale factor.

Finding Scale Factor of Similar Triangles

The image contains two similar right-angled triangles, Triangle CDE and Triangle VUW. We are given the lengths of the sides of each triangle, with side CD being 10 units, DE being 12 units, and CE being unknown but corresponding to side VW which is 9.6 units. We are asked to find the scale factor from Triangle CDE to Triangle VUW. The scale factor is the ratio of the lengths of corresponding sides in similar figures. To determine the scale factor between these two triangles, we can take the lengths of any pair of corresponding sides and divide them. Here, we can use DE and UW since those are the only two corresponding sides both of which we know the lengths. Since DE is the longer side in the larger triangle, we will divide the length of DE by the length of UW to get the scale factor. Let's do the calculation: Scale factor = (Length of DE in Triangle CDE) / (Length of UW in Triangle VUW) Scale factor = 12 / 9.6 When you divide 12 by 9.6, the result is: Scale factor = 1.25 This means that Triangle CDE is 1.25 times larger than Triangle VUW, or in other words, Triangle VUW is 1.25 times smaller than Triangle CDE. The scale factor is 1.25.

Determining Scale Factor of Similar Figures

The given image shows two similar figures (triangles), and we are asked to find the scale factor between them. To determine the scale factor, we compare the lengths of corresponding sides of the similar figures. From the image, we observe that side CD in the larger figure corresponds to side UV in the smaller figure. We can calculate the scale factor (k) by dividing the length of UV by the length of CD: k = UV / CD k = 9.6 / 12 To solve for k, we divide 9.6 by 12: k = 0.8 So, the scale factor between the two similar figures is 0.8.

Finding Scale Factor Between Two Figures

The image shows two similar figures, and we are tasked with finding the scale factor. To find the scale factor from one figure to another, you can divide the lengths of corresponding sides. In the image, you can use the lengths of sides DC and UV to determine the scale factor. DC is 5 units long, and UV is 4 units long, both representing the shortest sides of their respective figures. To find the scale factor from the larger figure (DC) to the smaller one (UV), divide the length of UV by the length of DC: Scale factor = UV / DC Scale factor = 4 / 5 Scale factor = 0.8 Therefore, the scale factor from figure DC to figure UV is 0.8.

Finding Scale Factor of Similar Triangles

The image shows two similar triangles. When two triangles are similar, their corresponding sides are proportional, meaning the ratio between the lengths of one pair of corresponding sides is the same as the ratio between the lengths of any other pair of corresponding sides. To find the scale factor from the larger triangle CDE to the smaller triangle VTU, we divide the lengths of one pair of corresponding sides from the two triangles. We can use sides DE and TU for this purpose: Scale factor = side TU / side DE Scale factor = 8 / 12 Scale factor = 2 / 3 So, the scale factor from triangle CDE to triangle VTU is 2/3.

Dilation of a Point

The question is asking for the image of the point (12, −4) after a dilation by a scale factor of \( \frac{1}{3} \), centered at the origin. To find the image of a point after dilation, you multiply the coordinates of the original point by the scale factor. Here the scale factor is \( \frac{1}{3} \). Thus, the transformed coordinates (x', y') of the original point (x, y) = (12, −4) would be calculated by: x' = x * scale factor y' = y * scale factor Calculating these: x' = 12 * \( \frac{1}{3} \) = 4 y' = −4 * \( \frac{1}{3} \) = −\( \frac{4}{3} \) Therefore, the image of the point (12, −4) after the dilation is (4, −\( \frac{4}{3} \)).

Dilation of a Point by Scale Factor of 3

The question asks for the image of the point (6, 12) after a dilation by a scale factor of 3, centered at the origin. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. When the center of dilation is the origin (0, 0), you can find the image of a point by multiplying the coordinates of the original point by the scale factor. Given the point (6, 12) and a scale factor of 3, the image is found by multiplying both coordinates by 3: Image of (6, 12) = (6 * 3, 12 * 3) = (18, 36). So, the image of the point (6, 12) after a dilation by a scale factor of 3, centered at the origin, is (18, 36).

Dilation of a Point with Scale Factor and Origin

To find the image of a point after a dilation with a scale factor, you multiply each coordinate of the original point by the scale factor. When the dilation is centered at the origin (0, 0), this operation is straightforward. The original point is (-6, 7), and the scale factor is 5. Therefore, you multiply each coordinate of the point by the scale factor: New x-coordinate: -6 * 5 = -30 New y-coordinate: 7 * 5 = 35 So the image of the point (-6, 7) after dilation by a scale factor of 5, centered at the origin, is (-30, 35).

Point Dilation by Scale Factor at Origin

To find the image of a point after a dilation by a scale factor centered at the origin, you can multiply each coordinate of the point by the scale factor. The given point is \((-4, -8)\), and the scale factor is \(\frac{1}{4}\). Therefore, the image of this point after dilation will be: \((-4 \times \frac{1}{4}, -8 \times \frac{1}{4}) = (-1, -2)\) So the image of the point \((-4, -8)\) after a dilation by a scale factor of \(\frac{1}{4}\) centered at the origin is \((-1, -2)\).