Example Question - proportional sides
Here are examples of questions we've helped users solve.
Geometric Similarity and Ratio Calculation in Triangles
<p>(a) In triangle ABM and CDM:</p> <p>\[\angle MAB = \angle MCD\] (Vertically opposite angles are equal)</p> <p>\[\angle AMB = \angle CMD\] (Each equals \(90^\circ\))</p> <p>Therefore, by AA (Angle-Angle) similarity criterion, \(\triangle ABM \sim \triangle CDM\).</p> <p>(b) The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides:</p> <p>\[\frac{{\text{area of } \triangle ABM}}{{\text{area of } \triangle CDM}} = \left(\frac{AM}{CM}\right)^2\]</p> <p>(c) \[\frac{{\text{area of } \triangle ABM}}{{\text{area of } \triangle ABC}} = \frac{\frac{1}{2} AM \times BM}{\frac{1}{2} AC \times BC}\]</p> <p>Given that \(BM = MC\) and \(AM = MC\),</p> <p>\[\frac{{\text{area of } \triangle ABM}}{{\text{area of } \triangle ABC}} = \left(\frac{MC}{AC}\right) \times \left(\frac{AM}{BC}\right)\]</p> <p>Since \(AB = AC\), \(BC = AB - AM\),</p> <p>\[\frac{{\text{area of } \triangle ABM}}{{\text{area of } \triangle ABC}} = \frac{MC}{AC} \times \frac{AM}{AB - AM}\]</p>
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.
Solving Triangle Ratios
The image shows two triangles with side lengths provided. The larger triangle has sides of length 21, 15, and 18, while the smaller triangle has one side labeled as 5 and two sides that are not labeled. It is stated that the ratio of the larger triangle to the smaller triangle is 3. To solve the problem, we'll assume that the corresponding sides of the triangles are proportional, meaning that the larger triangle is a scaled-up version of the smaller one. Since the ratio is 3, it implies that every length in the larger triangle is 3 times the length of the corresponding length in the smaller triangle. The smaller triangle has a side with a length of 5. To find the corresponding side on the larger triangle, we can divide one of the side lengths of the larger triangle by 3. Let's choose the side length of 15 for simplicity: 15 (larger side length) / 5 (smaller side length) = 3 This confirms that the side length of 5 in the smaller triangle corresponds to the side length of 15 in the larger triangle. To maintain the same scale, other sides of the smaller triangle should be 1/3 of their respective sides in the larger triangle. Thus, the other two sides of the smaller triangle can be found by dividing the remaining sides of the larger triangle by 3: 21 / 3 = 7 18 / 3 = 6 So the smaller triangle should have side lengths of 5, 7, and 6. However, since your question doesn't specify what needs to be solved, but only provides the ratio and the sizes of the triangles, I've provided you with the calculation that confirms the given ratio and the side lengths of the smaller triangle based on that ratio. If there's a specific question or calculation you need beyond this, please provide that information.
Calculating Missing Side of Similar Triangles
The image shows two triangles, and it states that the ratio of the larger triangle to the smaller triangle is 3. Since the ratio is given as the "size" or "scale" of the larger to the smaller, we should understand it to apply to corresponding sides of the triangles. Let's assume the triangles are similar; then the corresponding sides are proportional with the ratio of 3. We can use the given sides to find the unknown side of the smaller triangle. If we look at the larger triangle, the side shown is 21. We need to find what the corresponding side in the smaller triangle is. Since the ratio is 3, we divide the larger side by the ratio to find the corresponding smaller side: 21 (larger side) / 3 (ratio) = 7 (smaller corresponding side) So, the missing side of the smaller triangle is 7 units in length.
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.
Solving for Unknown Side Length in Similar Triangles
Unfortunately, the image provided does not clearly indicate all the numerical values necessary for solving the problem. It seems to show two similar triangles, where the lengths of corresponding sides are proportional, but I cannot see the value of the side that is labeled with an "x" in the smaller triangle. However, if we assume that the proportion between the sides of the two triangles visible (8 in the smaller triangle and 16 in the larger one) is consistent with the proportion of the unknown side (x) and its corresponding side in the larger triangle, we can find the value of "x". If the triangles are similar, then the ratio of a side of the smaller triangle to its corresponding side in the larger triangle should be the same for all sides. In this case, it seems that: 8 (smaller triangle) / 16 (larger triangle) = 1 / 2 Now apply this ratio to find "x" by using the corresponding side in the larger triangle (which is not visible in the image). If we had that length, we could calculate x as follows: x / corresponding side in the larger triangle = 1 / 2 Unfortunately, without the length of the corresponding side in the larger triangle, I cannot calculate the exact value of x. If you provide the length of the corresponding side in the larger triangle or any additional information, I would be able to solve for x. Please check the image for any additional information or clarify the lengths, and I'd be glad to assist you further.
Solving for Missing Side Length of Similar Triangles
The triangles in the image are similar, which means their corresponding sides are proportional. From the image, we can set up a ratio of the corresponding sides of the two similar triangles. Let's denote the unknown side length of the smaller triangle as \( x \). We are given the sides of the larger triangle as 8 and 16. Since the triangles are similar, the ratio of the sides of the smaller triangle to the larger one should be the same. So we have: \[\frac{x}{8} = \frac{7}{16}\] Now we can solve for \( x \) by cross-multiplying: \( 16x = 8 \times 7 \) \( 16x = 56 \) Now, divide both sides by 16 to find \( x \): \( x = \frac{56}{16} \) \( x = 3.5 \) Therefore, the missing side length of the smaller triangle is 3.5 units.
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