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Matching angles are equal
Matching sides are in the same ratio

","mcqOptions":[{"text":"

Matching angles are equal
Matching sides are in the same ratio

","correct":true,"explanation":"Equal corresponding angles give both shapes the same form, while proportional corresponding sides allow their sizes to differ by one consistent scale factor."},{"text":"

Matching angles are in the same ratio
Matching sides are equal

","correct":false,"explanation":"Similarity requires corresponding angles to be equal, not proportional, and corresponding sides to be in the same ratio. Requiring equal sides incorrectly restricts the shapes to congruent examples."},{"text":"

Matching angles are supplementary
Matching sides have the same difference

","correct":false,"explanation":"Supplementary angles add to $180°$ but need not match, and equal side differences do not preserve shape. Similarity requires equal corresponding angles and proportional corresponding sides."},{"text":"

Matching angles are equal
Matching sides have the same difference

","correct":false,"explanation":"Although equal corresponding angles are required, comparing side differences is an additive misconception. Corresponding side lengths must have one constant ratio, called the scale factor."}]},{"id":"d788fb86-d803-4dd3-afed-40c7bc1426ae","cardType":"cloze","points":null,"front":"The scale factor is the [ ] used to get from one shape to the similar shape.","back":"The scale factor is the multiplier used to get from one shape to the similar shape.","mcqOptions":[{"text":"multiplier","correct":true,"explanation":"The scale factor is the number by which every corresponding length is multiplied to transform one similar shape into the other."},{"text":"divisor","correct":false,"explanation":"A divisor may be used when reversing an enlargement, but it does not define the scale factor itself. The scale factor is the multiplier connecting corresponding lengths."},{"text":"difference","correct":false,"explanation":"Similar shapes are related multiplicatively rather than by adding a constant difference. The scale factor is the multiplier applied to every corresponding length."},{"text":"percentage","correct":false,"explanation":"A percentage can describe a relative change, but the scale factor is specifically the multiplier used to convert corresponding lengths from one shape to the other."}]},{"id":"d934fd6d-6a41-488d-8c14-c380a76abcee","cardType":"basic","points":1,"front":"Instead of using eyesight, how should you determine which sides of two triangles correspond?","back":"Use matching angles or the order of the letters in the triangle names.","mcqOptions":[{"text":"Use matching angles or the order of the letters in the triangle names.","correct":true,"explanation":"Corresponding vertices have equal angles and occupy matching positions in correctly ordered triangle names, so either method reliably identifies the corresponding sides."},{"text":"Use the positions on the page or the directions of the sides in the diagram.","correct":false,"explanation":"Similar triangles may be rotated or reflected, so page position and side direction can mislead. Use equal angles or the order of vertices in the triangle names."},{"text":"Use the matching side lengths or the size of the drawing in the diagram.","correct":false,"explanation":"Corresponding sides need not have equal lengths, and diagrams may not be drawn to scale. Match equal angles or use the ordered letters in the triangle names."},{"text":"Use the matching line thicknesses or the order of the side lengths in the diagram.","correct":false,"explanation":"Line thickness has no geometric meaning, and estimated length order can be unreliable. Corresponding sides should be found from matching angles or ordered triangle names."}]},{"id":"87a6bc0b-8b47-4b2d-8e4c-b41e12b067ac","cardType":"cloze","points":null,"front":"To find a missing length on a larger shape, [ ] the corresponding smaller length by the scale factor.","back":"To find a missing length on a larger shape, multiply the corresponding smaller length by the scale factor.","mcqOptions":[{"text":"multiply","correct":true,"explanation":"The larger shape’s corresponding length is found by multiplying the smaller length by the scale factor, since an enlargement increases every length by the same ratio."},{"text":"divide","correct":false,"explanation":"Dividing reverses an enlargement and would find the smaller corresponding length. To move from the smaller shape to the larger shape, multiply by the scale factor."},{"text":"add","correct":false,"explanation":"Adding the scale factor treats enlargement as an additive change, but similar lengths change proportionally. Multiply the smaller length by the scale factor instead."},{"text":"subtract","correct":false,"explanation":"Subtracting would reduce the length and does not preserve proportionality. For the larger similar shape, multiply the corresponding smaller length by the scale factor."}]},{"id":"482115c7-2d2f-4aea-a084-0952b4e4b075","cardType":"cloze","points":null,"front":"To find a missing length on a smaller shape, [ ] the corresponding larger length by the scale factor.","back":"To find a missing length on a smaller shape, divide the corresponding larger length by the scale factor.","mcqOptions":[{"text":"divide","correct":true,"explanation":"Moving from the larger shape to the smaller shape reverses the enlargement, so the larger corresponding length is divided by the scale factor."},{"text":"multiply","correct":false,"explanation":"Multiplication moves from the smaller shape to the larger one. To reverse that enlargement and find the smaller corresponding length, divide by the scale factor."},{"text":"add","correct":false,"explanation":"Adding the scale factor assumes a fixed increase rather than a proportional relationship. Find the smaller length by dividing the larger length by the scale factor."},{"text":"subtract","correct":false,"explanation":"Subtracting the scale factor is an additive shortcut that does not preserve similarity. The enlargement is reversed by dividing the larger length by the scale factor."}]},{"id":"5ca1e00b-3c57-4c05-bd8f-ee7aa0e4155c","cardType":"basic","points":1,"front":"A small triangle is inside a large triangle. What geometric feature ensures they are similar?","back":"A side of the small triangle is parallel to a side of the large triangle.","mcqOptions":[{"text":"A side of the small triangle is parallel to a side of the large triangle.","correct":true,"explanation":"Parallel sides create equal corresponding angles, while the triangles also share the angle at the common vertex. Two equal angle pairs establish similarity by angle-angle."},{"text":"A side of the small triangle is perpendicular to a side of the large triangle.","correct":false,"explanation":"Perpendicular sides create a right angle between the lines but do not establish equal corresponding angles. Parallel sides produce the angle relationships needed for triangle similarity."},{"text":"A side of the small triangle is equal to a side of the large triangle.","correct":false,"explanation":"One pair of equal sides is insufficient to prove similarity and may occur in unrelated triangles. A parallel pair of sides establishes the required matching angles."},{"text":"A side of the small triangle is bisected by a side of the large triangle.","correct":false,"explanation":"Bisecting a side concerns equal segments and does not guarantee matching angles or proportional sides. The relevant feature is that one pair of corresponding sides is parallel."}]},{"id":"c76ae69b-602a-4114-bcf0-c77dadbced1f","cardType":"cloze","points":null,"front":"In nested triangles $ABE$ and $ACD$ where $BE \\parallel CD$, the ratio of the sides is: $$\\frac{AB}{AC} = \\frac{AE}{AD} = \\frac{\\text{[...]}}{CD}$$","back":"In nested triangles $ABE$ and $ACD$ where $BE \\parallel CD$, the ratio of the sides is: $$\\frac{AB}{AC} = \\frac{AE}{AD} = \\frac{\\boxed{BE}}{CD}$$","mcqOptions":[{"text":"$$BE$","correct":true,"explanation":"Side $BE$ of the smaller triangle corresponds to side $CD$ of the larger triangle because both lie opposite the shared vertex $A$ and are parallel."},{"text":"$$CE$","correct":false,"explanation":"The segment $CE$ is not a side of the smaller triangle $ABE$, so it cannot be its numerator. The side corresponding to $CD$ is $BE$."},{"text":"$$DE$","correct":false,"explanation":"The segment $DE$ joins points on different sides of the nested construction and is not a corresponding triangle side. The smaller side parallel to $CD$ is $BE$."},{"text":"$$AB$","correct":false,"explanation":"Side $AB$ corresponds to $AC$, as already shown by the first ratio. Since $CD$ corresponds to the parallel side of the smaller triangle, the numerator is $BE$."}]},{"id":"412bee53-6a67-418b-9998-10eddd83adfe","cardType":"cloze","points":null,"front":"In nested triangles $ABE$ and $ACD$ where $BE \\parallel CD$, the ratio of the sides is: $$\\frac{AB}{AC} = \\frac{AE}{AD} = \\frac{BE}{\\text{[...]}}$$","back":"In nested triangles $ABE$ and $ACD$ where $BE \\parallel CD$, the ratio of the sides is: $$\\frac{AB}{AC} = \\frac{AE}{AD} = \\frac{BE}{\\boxed{CD}}$$","mcqOptions":[{"text":"$$CD$","correct":true,"explanation":"Side $CD$ of the larger triangle corresponds to $BE$ of the smaller triangle because the sides are parallel and join matching vertices."},{"text":"$$AD$","correct":false,"explanation":"Side $AD$ corresponds to $AE$, as shown by the middle ratio, rather than to $BE$. The larger side corresponding to $BE$ is $CD$."},{"text":"$$AC$","correct":false,"explanation":"Side $AC$ corresponds to $AB$, as shown by the first ratio. The side paired with $BE$ must instead be the parallel side $CD$."},{"text":"$$BE$","correct":false,"explanation":"Using $BE$ in both numerator and denominator would always give $1$, ignoring the scale difference. Its corresponding side in the larger triangle is $CD$."}]},{"id":"2113f968-7fb4-4897-8ad3-94e51f9f8d62","cardType":"basic","points":2,"front":"When calculating the scale factor for nested triangles (one inside another), what must you do if a side is split into two parts (e.g., $AB$ and $BC$)?","back":"Add the two parts together to find the full side length of the larger triangle.","mcqOptions":[{"text":"Add the two parts together to find the full side length of the larger triangle.","correct":true,"explanation":"The larger triangle’s side consists of both adjacent segments, so their lengths must be added before it can be compared with the corresponding side of the smaller triangle."},{"text":"Subtract the smaller part from the larger part to find the full side length of the larger triangle.","correct":false,"explanation":"Subtraction finds a difference rather than the complete side. The two adjacent parts must be added because together they form the full side of the larger triangle."},{"text":"Use only the longer part as the full side length of the larger triangle.","correct":false,"explanation":"Using one segment ignores the other section of the larger triangle’s side. Add both adjacent parts to obtain the full corresponding length before finding the scale factor."},{"text":"Multiply the two parts together to find the full side length of the larger triangle.","correct":false,"explanation":"Multiplying lengths produces square units and cannot give the length of the complete side. Add the two adjacent segments to find the larger triangle’s full side length."}]},{"id":"aecc277e-e0cf-4d15-bd14-6a0e5e3f9cbc","cardType":"basic","points":1,"front":"In a bow-tie diagram where two lines cross at $P$, which sides typically correspond?","back":"Sides that lie on the same straight line passing through $P$.","mcqOptions":[{"text":"Sides that lie on the same straight line passing through $P$.","correct":true,"explanation":"Each side of one triangle corresponds to the side of the other triangle continuing along the same intersecting line through $P$, even though they point in opposite directions."},{"text":"Sides that lie on parallel straight lines on opposite sides of $P$.","correct":false,"explanation":"The parallel sides are usually the triangles’ bases, not the paired sides passing through the crossing. Corresponding sloping sides lie on the same straight line through $P$."},{"text":"Sides that have the same numerical length on opposite sides of $P$.","correct":false,"explanation":"Corresponding sides of similar triangles are proportional, but they need not have equal lengths. In a bow-tie diagram, correspondence is identified by sides lying on the same line through $P$."},{"text":"Sides that meet at the same parallel base away from $P$.","correct":false,"explanation":"The triangles have separate parallel bases, so their sides do not meet at one common base. Corresponding sloping sides are those on the same straight line through $P$."}]},{"id":"77a39892-fabc-4311-a9bb-cef8206a987a","cardType":"basic","points":2,"front":"Why are triangles in a bow-tie diagram with parallel bases mathematically similar?","back":"

Vertically opposite angles at the crossing point are equal
Parallel lines create equal corresponding (or alternate) angles

","mcqOptions":[{"text":"

Vertically opposite angles at the crossing point are equal
Parallel lines create equal corresponding (or alternate) angles

","correct":true,"explanation":"These two angle relationships provide equal pairs of angles in the triangles. Therefore, the triangles satisfy the angle-angle similarity condition and must be mathematically similar."},{"text":"

Adjacent angles at the crossing point are equal
Parallel lines create equal co-interior angles

","correct":false,"explanation":"Adjacent angles at an intersection are generally supplementary, while co-interior angles between parallel lines sum to $180^\\circ$. Similarity follows from vertically opposite and corresponding or alternate angles being equal."},{"text":"

Vertically opposite sides at the crossing point are equal
Parallel lines create equal side lengths in both triangles

","correct":false,"explanation":"Vertically opposite describes angles, not sides, and parallel lines do not guarantee equal lengths. The triangles are similar because vertically opposite and corresponding or alternate angles are equal."},{"text":"

Supplementary angles at the crossing point are equal
Parallel lines create equal opposite angles

","correct":false,"explanation":"Supplementary angles add to $180^\\circ$ but are not necessarily equal, and “opposite angles” is not the relevant parallel-line rule. Use vertically opposite and corresponding or alternate angles."}]},{"id":"d1f99d9d-bde1-4372-8f79-750cf1e11d7d","cardType":"cloze","points":null,"front":"If the ratio of two corresponding sides $AB:AC$ is $2:5$, the scale factor to get from $AB$ to $AC$ is $\\text{[...]}$.","back":"If the ratio of two corresponding sides $AB:AC$ is $2:5$, the scale factor to get from $AB$ to $AC$ is $\\boxed{2.5}$.","mcqOptions":[{"text":"$$2.5$","correct":true,"explanation":"The scale factor from $AB$ to $AC$ is the target-to-original ratio, $5 \\div 2 = 2.5$. Multiplying the length of $AB$ by $2.5$ gives $AC$."},{"text":"$$0.4$","correct":false,"explanation":"This comes from reversing the division and calculating $2 \\div 5$. That is the scale factor from $AC$ to $AB$; from $AB$ to $AC$, use $5 \\div 2 = 2.5$."},{"text":"$$1.5$","correct":false,"explanation":"This incorrectly uses the difference between the ratio numbers rather than dividing corresponding lengths. The required scale factor is $5 \\div 2 = 2.5$."},{"text":"$$3.5$","correct":false,"explanation":"Adding or otherwise combining the ratio numbers does not produce a scale factor. Divide the target ratio part by the original part: $5 \\div 2 = 2.5$."}]},{"id":"2c99684f-7554-4553-8237-b28ab285f430","cardType":"basic","points":1,"front":"If you know the total length $AD$ and the small length $AE$, how do you find the length of the remaining segment $ED$?","back":"$$$ED = AD - AE$$ (Subtract the smaller triangle side from the larger triangle side).","mcqOptions":[{"text":"$$$ED = AD - AE$$ (Subtract the smaller triangle side from the larger triangle side).","correct":true,"explanation":"Since $AD$ is the complete length and consists of $AE$ plus $ED$, rearranging $AD = AE + ED$ gives $ED = AD - AE$."},{"text":"$$$ED = AD + AE$$ (Add the smaller triangle side to the larger triangle side).","correct":false,"explanation":"Adding $AE$ to the total $AD$ counts that section twice and produces a length greater than the whole side. The remaining segment is $ED = AD - AE$."},{"text":"$$$ED = AE - AD$$ (Subtract the larger triangle side from the smaller triangle side).","correct":false,"explanation":"Reversing the subtraction would give a negative value because $AD$ is the complete, longer side. Subtract the known small segment from the total: $ED = AD - AE$."},{"text":"$$$ED = AD \\div AE$$ (Divide the larger triangle side by the smaller triangle side).","correct":false,"explanation":"Division gives a dimensionless ratio or scale factor, not the length of the missing segment. Remove the known part from the total using $ED = AD - AE$."}]},{"id":"8e077481-f0ce-464b-9cca-58e84f1ad8cc","cardType":"basic","points":2,"front":"Why are all rectangles not necessarily similar, even though their angles are all $90^\\circ$?","back":"Their length-to-width ratios (or the scale factors for length and width) might be different.","mcqOptions":[{"text":"Their length-to-width ratios (or the scale factors for length and width) might be different.","correct":true,"explanation":"Similarity requires corresponding side lengths to share one scale factor. Rectangles all have equal corresponding angles, but differing length-to-width ratios mean their sides are not proportional."},{"text":"Their opposite side lengths (or the pairs of parallel sides in each shape) might be different.","correct":false,"explanation":"Similar shapes may have different actual side lengths, so this alone does not prevent similarity. Rectangles are dissimilar when their length-to-width ratios, or corresponding scale factors, differ."},{"text":"Their total interior angles (or the number of right angles in each shape) might be different.","correct":false,"explanation":"Every rectangle has four right angles and an interior-angle sum of $360^\\circ$, so these cannot distinguish similarity. The deciding condition is whether corresponding side lengths are proportional."},{"text":"Their areas and perimeters (or the overall sizes of the two shapes) might be different.","correct":false,"explanation":"Similar rectangles can have different areas and perimeters because similarity does not require equal size. They fail to be similar only when their length-to-width ratios are different."}]},{"id":"c6552d9b-6e2f-49f7-8a02-5395fe69987f","cardType":"cloze","points":null,"front":"To show two rectangles are not similar, show that the scale factor for the [ ] is different from the scale factor for the widths.","back":"To show two rectangles are not similar, show that the scale factor for the lengths is different from the scale factor for the widths.","mcqOptions":[{"text":"lengths","correct":true,"explanation":"A rectangle’s length and width are corresponding linear dimensions. If they require different scale factors, the sides are not proportional and the two rectangles cannot be similar."},{"text":"areas","correct":false,"explanation":"Area uses a squared scale factor, so it should not be compared directly with a width’s linear scale factor. Compare the scale factors for the lengths and widths instead."},{"text":"angles","correct":false,"explanation":"Angles do not have a linear scale factor, and all rectangles already have four $90^\\circ$ angles. Compare the scale factor for the lengths with that for the widths."},{"text":"volumes","correct":false,"explanation":"A rectangle is two-dimensional and has no volume, so a volume scale factor is not relevant. To test similarity, compare the linear scale factors for its lengths and widths."}]},{"id":"ab2db1ea-9a01-4b83-bf40-794dc9df263c","cardType":"cloze","points":null,"front":"To show two rectangles are not similar, show that the scale factor for the lengths is different from the scale factor for the [ ].","back":"To show two rectangles are not similar, show that the scale factor for the lengths is different from the scale factor for the widths.","mcqOptions":[{"text":"widths","correct":true,"explanation":"For similar rectangles, corresponding lengths and widths must be enlarged or reduced by the same linear scale factor. Different factors show that the side proportions have changed."},{"text":"areas","correct":false,"explanation":"Area changes by the square of the linear scale factor, so its factor should not normally equal the length factor. Compare the scale factors for lengths and widths instead."},{"text":"angles","correct":false,"explanation":"All rectangles have the same four right angles, and angles are not multiplied by a linear scale factor. Non-similarity is shown by different scale factors for lengths and widths."},{"text":"volumes","correct":false,"explanation":"Rectangles are two-dimensional, so volume is not a relevant measurement or scale factor. The correct comparison is between the linear scale factors for lengths and widths."}]},{"id":"be52d227-31d6-4edd-82c8-ebea26dcdd77","cardType":"basic","points":1,"front":"If the scale factor from triangle $A$ to triangle $B$ is $1.5$, and a side on triangle $B$ is $24$ cm, how do you find the matching side on triangle $A$?","back":"$$$24 \\div 1.5 = 16\\text{ cm}$$","mcqOptions":[{"text":"$$$24 \\div 1.5 = 16\\text{ cm}$$","correct":true,"explanation":"The scale factor from $A$ to $B$ is $1.5$, so reversing the enlargement requires division by $1.5$. Therefore, the matching side on triangle $A$ is $16$ cm."},{"text":"$$$24 \\times 1.5 = 36\\text{ cm}$$","correct":false,"explanation":"Multiplying applies the scale factor from triangle $A$ to triangle $B$, but the question asks for the reverse direction. Dividing $24$ by $1.5$ gives the correct length of $16$ cm."},{"text":"$$$24 \\div 1.5 = 18\\text{ cm}$$","correct":false,"explanation":"The correct operation has been selected, but the division is inaccurate. Since $16 \\times 1.5 = 24$, the matching side is $16$ cm, not $18$ cm."},{"text":"$$$24 - 1.5 = 22.5\\text{ cm}$$","correct":false,"explanation":"A scale factor represents multiplication, not an amount to subtract from a length. Reverse the enlargement by calculating $24 \\div 1.5$, which gives $16$ cm."}]},{"id":"2607239c-4032-43fa-be00-adc9f7107ea9","cardType":"cloze","points":null,"front":"Arrow marks on a geometry diagram usually indicate that lines are [ ].","back":"Arrow marks on a geometry diagram usually indicate that lines are parallel.","mcqOptions":[{"text":"parallel","correct":true,"explanation":"Matching arrow marks show that lines have the same direction and never meet, so the lines are parallel. Equal lengths are instead indicated by matching tick marks."},{"text":"perpendicular","correct":false,"explanation":"Perpendicular lines are normally identified by a small right-angle square where they meet. Matching arrow marks indicate parallel lines, which remain the same distance apart."},{"text":"equal","correct":false,"explanation":"Equal line lengths are shown using matching short tick marks across the segments. Arrow marks placed on lines indicate that those lines are parallel."},{"text":"intersecting","correct":false,"explanation":"Intersecting lines cross at a point and do not require matching arrow notation. Arrow marks specifically identify lines that are parallel and therefore do not meet."}]},{"id":"d6217d5e-4ea0-49e5-8a96-648322c0cf09","cardType":"basic","points":1,"front":"What is the scale factor from a length of $140$ mm to a length of $180$ mm?","back":"$$$\\frac{180}{140} = \\frac{9}{7} \\approx 1.29$$","mcqOptions":[{"text":"$$$\\frac{180}{140} = \\frac{9}{7} \\approx 1.29$$","correct":true,"explanation":"The scale factor is the new length divided by the original length. Thus $180 \\div 140$ simplifies to $9/7$, which is approximately $1.29$."},{"text":"$$$\\frac{140}{180} = \\frac{7}{9} \\approx 0.78$$","correct":false,"explanation":"This reverses the division and finds the scale factor from $180$ mm to $140$ mm. From $140$ mm to $180$ mm, calculate $180/140 = 9/7 \\approx 1.29$."},{"text":"$$$\\frac{180}{140} = \\frac{8}{7} \\approx 1.14$$","correct":false,"explanation":"Dividing both numerator and denominator by $20$ gives $9/7$, not $8/7$. Therefore, the scale factor is approximately $1.29$, rather than $1.14$."},{"text":"$$$\\frac{180}{140} = \\frac{9}{5} = 1.8$$","correct":false,"explanation":"The fraction has been simplified inconsistently, because numerator and denominator must be divided by the same number. Dividing both by $20$ gives $9/7 \\approx 1.29$."}]},{"id":"ee7c6c1d-d6a1-4e41-a4d0-06f4a1cc66ae","cardType":"cloze","points":null,"front":"A [ ] tells you the comparison between lengths, but you must use actual lengths to find missing side values.","back":"A ratio tells you the comparison between lengths, but you must use actual lengths to find missing side values.","mcqOptions":[{"text":"ratio","correct":true,"explanation":"A ratio compares corresponding lengths and shows their relative sizes. Once the ratio is known, it can be applied to an actual given length to calculate a missing side."},{"text":"angle","correct":false,"explanation":"An angle describes rotation or the space between two lines, not a comparison between side lengths. A ratio provides the required comparison between corresponding lengths."},{"text":"perimeter","correct":false,"explanation":"A perimeter is the total distance around a shape rather than a comparison between lengths. The comparison needed for corresponding sides is expressed as a ratio."},{"text":"bearing","correct":false,"explanation":"A bearing gives a direction measured clockwise from north, so it does not compare corresponding side lengths. That comparison is represented by a ratio."}]},{"id":"f738d644-2042-4eac-b0cc-6548fda80de2","cardType":"cloze","points":null,"front":"A ratio tells you the comparison between lengths, but you must use [ ] to find missing side values.","back":"A ratio tells you the comparison between lengths, but you must use actual lengths to find missing side values.","mcqOptions":[{"text":"actual lengths","correct":true,"explanation":"A ratio gives only relative sizes. To calculate a numerical missing side, apply that ratio or its scale factor to a known actual length."},{"text":"angle measurements","correct":false,"explanation":"Similar shapes have equal corresponding angles, so angle measurements do not determine the numerical size of a missing side. A known actual length is needed with the ratio."},{"text":"area ratios","correct":false,"explanation":"An area ratio compares two-dimensional quantities and is the square of the length scale factor. To find a missing side directly, use the length ratio with actual lengths."},{"text":"visual estimates","correct":false,"explanation":"Geometry diagrams may not be drawn accurately to scale, so estimating from appearance is unreliable. Missing sides must be calculated from the ratio and actual given lengths."}]},{"id":"2a2a78de-a18d-44f7-97ec-ff3562acf2b1","cardType":"basic","points":1,"front":"In back-to-back triangles meeting at point $C$ with $AB \\parallel DE$, if $AB=8$ and $DE=12$, what is the scale factor from the small triangle to the large one?","back":"$$$\\frac{12}{8} = 1.5$$","mcqOptions":[{"text":"$$$\\frac{12}{8} = 1.5$$","correct":true,"explanation":"The scale factor from the small triangle to the large triangle is corresponding large length divided by small length. Therefore, $12 \\div 8 = 1.5$."},{"text":"$$$\\frac{8}{12} = \\frac{2}{3}$$","correct":false,"explanation":"This division gives the scale factor from the large triangle to the small triangle, reversing the requested direction. Small to large requires $12/8 = 1.5$."},{"text":"$$$\\frac{12}{8} = 2$$","correct":false,"explanation":"The correct corresponding lengths have been divided in the correct order, but the arithmetic is wrong. Since $12$ is one and a half times $8$, the factor is $1.5$."},{"text":"$$$\\frac{12-8}{8} = 0.5$$","correct":false,"explanation":"This calculates the proportional increase of $0.5$, not the enlargement scale factor. The scale factor compares the complete lengths, giving $12/8 = 1.5$."}]},{"id":"445d4eca-5afd-40c3-9ff7-170047228172","cardType":"basic","points":2,"front":"List the first two steps for solving a similar shapes problem in an exam.","back":"

Mark corresponding sides on the diagram
Calculate one scale factor using a known pair of matching sides

","mcqOptions":[{"text":"

Mark corresponding sides on the diagram
Calculate one scale factor using a known pair of matching sides

","correct":true,"explanation":"Identifying corresponding sides prevents mismatched comparisons. Dividing a known pair of matching lengths then gives the scale factor needed to calculate other corresponding sides."},{"text":"

Mark corresponding angles on the diagram
Calculate one scale factor using a known pair of matching angles

","correct":false,"explanation":"Corresponding angles are equal in similar shapes, so their ratio is always $1$ and cannot give the length scale factor. Mark corresponding sides and compare matching lengths instead."},{"text":"

Measure corresponding sides on the diagram
Calculate one scale factor using the measured pair of matching sides

","correct":false,"explanation":"Exam diagrams are not necessarily drawn to scale, so measured lengths may be inaccurate. Use the stated side lengths after marking which sides correspond."},{"text":"

Mark corresponding sides on the diagram
Calculate one scale factor using a known pair of non-matching sides

","correct":false,"explanation":"Comparing non-matching sides produces an invalid scale factor because they occupy different positions in the shapes. The known lengths must be a pair of corresponding sides."}]}]}],"examCountdown":null,"heading":"Similar Shapes (Lengths)","title":"Similar Shapes (Lengths) Flashcards","topbar":{"backHref":"/igcse/maths/cie/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"866d3406-cdbe-4884-9dea-f974bf13375b","walled":"$L39"}],"$L3a","$L3b"]