Example Question - similar triangles
Here are examples of questions we've helped users solve.
Exploring Similarity and Area Ratios in Geometric Figures
<p>(a) Triangle \( AXM \) is similar to triangle \( CXD \) because they have two pairs of corresponding angles that are equal: \( \angle AXM = \angle CXD \) (both are right angles), and \( \angle AMX = \angle CDX \) (they are vertical angles). Therefore, by AA (Angle-Angle) similarity criterion, \( \triangle AXM \sim \triangle CXD \).</p> <p>(b) To find the value of \( \frac{\text{area of } \triangle AMX}{\text{area of } \triangle CXD} \), we can use the fact that the ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. If \( AM = x \) and \( CD = kx \) for some proportional constant \( k \), then:</p> <p>\[ \frac{\text{area of } \triangle AMX}{\text{area of } \triangle CXD} = \left( \frac{x}{kx} \right)^2 = \left( \frac{1}{k} \right)^2 \]</p> <p>(c) By the same principle, to find the value of \( \frac{\text{area of } \triangle AXM}{\text{area of } \triangle ACDX} \), we add the areas of \( \triangle AMX \) and \( \triangle CXD \) to get the area of \( \triangle ACDX \). Assuming \( AM = x \) and \( CD = kx \), the area of \( \triangle ACDX \) is the sum of the areas of \( \triangle AMX \) and \( \triangle CXD \):</p> <p>\[ \text{area of }\triangle ACDX = \text{area of }\triangle AMX + \text{area of }\triangle CXD \]</p> <p>\[ \frac{\text{area of }\triangle AXM}{\text{area of }\triangle ACDX} = \frac{\text{area of }\triangle AXM}{\text{area of }\triangle AXM + \text{area of }\triangle CXD} \]</p> <p>\[ \frac{\text{area of }\triangle AXM}{\text{area of }\triangle ACDX} = \frac{x^2}{x^2 + (kx)^2} \]</p> <p>Without specific values for AM and CD, we can't simplify further. If more information is provided, then the ratio can be expressed in simplest form.</p>
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>
Similarity and Ratio in a Parallelogram
<p>\text{(a) Similarity Reasoning:}</p> <p>Triangle \triangle AXM \:\text{is similar to}\: \triangle CXD \:\text{because}:</p> <p>\text{1. } \angle AXM = \angle CXD \:\text{opposite angles are equal in a parallelogram}.</p> <p>\text{2. } \angle A = \angle C \:\text{alternate angles are equal as AD} \parallel \text{BC in a parallelogram}.</p> <p>\text{Therefore, by AA similarity criterion, } \triangle AXM \sim \triangle CXD</p> <p>\text{(b) Area Ratio Calculation:}</p> <p>\text{Since } \triangle AXM \sim \triangle CXD,\: \text{the ratio of their areas is the square of the ratio of their corresponding sides.}</p> <p>\text{The corresponding sides are } AM \:\text{and} \: CD.\:</p> <p>AM = \frac{1}{2}AD \:\text{since M is the midpoint of AD}.</p> <p>CD = AD \:\text{as opposite sides of a parallelogram are equal}.</p> <p>\text{Therefore, the ratio of } AM \:\text{to} \: CD \:\text{is} \:\frac{AM}{CD} = \frac{1/2 \cdot AD}{AD} = \frac{1}{2}.</p> <p>\text{The ratio of areas is } (\frac{AM}{CD})^2 = (\frac{1}{2})^2 = \frac{1}{4}.</p> <p>\text{Hence,} \:\frac{\text{area of } \triangle AXM}{\text{area of } \triangle CXD} = \frac{1}{4}.</p>
Analysis of Geometric Similarity and Area Ratio in a Quadrilateral
(a) <p>\angle AXM = \angle CXD \quad (\text{vertically opposite angles are equal})</p> <p>\angle AMX = \angle CDX \quad (\text{corresponding angles of parallel lines are equal})</p> <p>\angle A = \angle C \quad (\text{given})</p> <p>\text{By AA similarity criterion,} \triangle AXM \sim \triangle CXD.</p> (b) <p>\frac{\text{area of }\triangle AMX}{\text{area of }\triangle CXD} = \left(\frac{AM}{CD}\right)^2</p> <p>\text{Since }\triangle AXM \sim \triangle CXD\text{, their sides are proportional. Therefore, } \frac{AM}{CD} = \frac{AX}{CX}</p> <p>\text{Therefore, the area ratio is } \left(\frac{AX}{CX}\right)^2.</p>
Solving for the Sine of an Angle in Similar Triangles
<p>Given that \(\sin(F) = \frac{308}{317}\) and triangles FGH and JKL are similar with angle F corresponding to angle J, and angles G and K are right angles, we have:</p> <p>\(\sin(J) = \sin(F) = \frac{308}{317}\)</p> <p>Therefore, the value of \(\sin(J)\) is \(\frac{308}{317}\).</p> <p>The correct option is C) \(\frac{308}{317}\).</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.
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 Between Two Triangles
The image shows two similar triangles, triangle CDE and triangle VUW, with corresponding side lengths marked. Side CD is 10 units, side DE is 12 units, and side UW is given as 9.6 units. We are asked to find the scale factor. To find the scale factor between the two triangles, we compare the corresponding sides. Since the image shows side DE of triangle CDE corresponding to side UW of triangle VUW, and Triangle VUW is the smaller one, we can determine the scale factor by dividing the length of side UW by the length of side DE. Scale Factor = Length of side UW / Length of side DE Plugging in the given values: Scale Factor = 9.6 / 12 Scale Factor = 0.8 Therefore, the scale factor is 0.8.
Solving Triangles with Given Ratios
The image shows two triangles, with a given statement that "The ratio of the LARGER triangle to the SMALLER triangle is \( \frac{3}{5} \)." To use this information to solve the problem, we must understand that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. In this case, the larger triangle has sides of lengths 21 and 18, but we are not given the corresponding sides of the smaller triangle. However, we do know one of the sides of the smaller triangle, which is 5. Since we know the ratio of the areas is \( \frac{3}{5} \), we can set up an equation using the side of the smaller triangle that we know: \[ \left( \frac{\text{Side of larger triangle}}{\text{Side of smaller triangle}} \right)^2 = \frac{3}{5} \] We can use either 21 (corresponding to the larger side) or 18 (corresponding to the smaller side) for the "Side of larger triangle" in the equation. Let's use 18 as it corresponds with the side length of 5 in the smaller triangle: \[ \left( \frac{18}{5} \right)^2 = \frac{3}{5} \] To find the "Side of larger triangle" that corresponds to the smaller triangle's side of length 5, we can use the square root to get: \[ \left( \frac{18}{5} \right) = \sqrt{\frac{3}{5}} \] \[ \frac{18}{5} = \frac{\sqrt{3}}{\sqrt{5}} \] Now, to solve for the corresponding side in the larger triangle, we cross-multiply: \[ 5\sqrt{3} = 18\sqrt{5} \] \[ \text{Side of larger triangle} = \frac{18\sqrt{5}}{\sqrt{3}} \] \[ \text{Side of larger triangle} = \frac{18\sqrt{3}\sqrt{5}}{3} \] \[ \text{Side of larger triangle} = 6\sqrt{15} \] So the side of the larger triangle that corresponds to the side of length 5 in the smaller triangle has a length of \( 6\sqrt{15} \), which would be the correct value to fill in the blank space in the image.
Find Missing Base Length of Larger Triangle with Given Ratio
The image shows two triangles, one larger and one smaller, with the ratio of the larger triangle to the smaller triangle being given as 3. The larger triangle has side lengths of 18, 21, and the base is not labeled. The smaller triangle has side lengths of 5 and 6, with the base also not labeled. To find the missing base length of the larger triangle, we need to use the given ratio which applies to corresponding sides of similar triangles. We can see that the larger triangle's sides are 3 times longer than the corresponding sides of the smaller one (because 15 x 3 = 45 and 18 x 3 = 54, with 18 being analogous to 6, so we can find the analogous side to 5 in the smaller triangle by multiplying 5 by 3). Given the ratio is 3, and one side of the smaller triangle is 5, the corresponding side on the larger triangle should be 5 x 3 = 15. Therefore, the base of the larger triangle is 15 units long.
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.
Calculating Scale Factor Between Two Triangles
The image shows two similar triangles with corresponding side lengths given. To find the scale factor, we can divide the length of a side of one triangle by the corresponding side length of the other triangle. From triangle CDE: CD = 10 DE = 12 CE = 5 From triangle VUT: VU = 9.6 UT = 8 VT = ? (we don't know this length, and we don't need it to find the scale factor). Let's use the sides CD and VU for our calculation since both of these corresponding sides are given: VU (from triangle VUT) / CD (from triangle CDE) = 9.6 / 10 = 0.96 Therefore, the scale factor from triangle CDE to triangle VUT is 0.96.
Finding Missing Side Length in Similar Triangles
The image shows two similar triangles, and we're asked to find the missing side length, labeled as "x". When two triangles are similar, corresponding side lengths are proportional. In the image, the smaller triangle has sides of length 4 and "x", and the larger triangle has corresponding sides of length 8 and 16. We can set up a proportion using the known side lengths of the triangles: \[\frac{x}{16} = \frac{4}{8}\] To find x, we solve for it by cross-multiplying: \[8x = 4 \times 16\] \[8x = 64\] \[x = \frac{64}{8}\] \[x = 8\] Therefore, the missing side length "x" is 8 units long.
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.
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