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it lies opposite the right angle and is always the triangle’s longest side."},{"text":"The adjacent side (the side next to the right angle).","correct":false,"explanation":"The adjacent side depends on which acute angle is being considered, so it is not necessarily $c$. In Pythagoras, $c$ must be the hypotenuse opposite the right angle."},{"text":"The shortest side (the side next to the right angle).","correct":false,"explanation":"Choosing the shortest side confuses the fixed role of $c$ with relative side lengths. The correct side is the hypotenuse, which is opposite the right angle and longest."},{"text":"The opposite side (the side opposite an acute angle).","correct":false,"explanation":"The opposite side in trigonometry changes according to the chosen acute angle. In Pythagoras, $c$ specifically represents the hypotenuse opposite the right angle."}]},{"id":"bdea9465-8500-4119-b3a2-9e4d63cdd092","cardType":"cloze","points":null,"front":"A [ ] joins two opposite corners of a 3D shape, passing through the inside of the solid.","back":"A space diagonal joins two opposite corners of a 3D shape, passing through the inside of the solid.","mcqOptions":[{"text":"space diagonal","correct":true,"explanation":"A space diagonal connects opposite vertices while travelling through the interior of a three-dimensional solid, rather than remaining on one of its faces."},{"text":"face diagonal","correct":false,"explanation":"A face diagonal joins opposite corners of one two-dimensional face and stays on the solid’s surface. A diagonal passing through the interior is a space diagonal."},{"text":"slant height","correct":false,"explanation":"A slant height measures a sloping surface length, commonly on a cone or pyramid; it does not generally connect opposite vertices. The required term is space diagonal."},{"text":"base edge","correct":false,"explanation":"A base edge joins two neighbouring vertices and lies along the boundary of the base. A line joining opposite corners through the solid is a space diagonal."}]},{"id":"174b45ce-3520-4fc3-b781-73624efd8245","cardType":"basic","points":2,"front":"What is the 2-step strategy for finding a cuboid's space diagonal without using the shortcut formula?","back":"

Find the face/base diagonal using Pythagoras.
Use that diagonal and the height in a second Pythagoras calculation.

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

Find the face/base diagonal using Pythagoras.
Use that diagonal and the height in a second Pythagoras calculation.

","correct":true,"explanation":"The base diagonal forms the first right-angled triangle. It then becomes one perpendicular side of a second right-angled triangle whose hypotenuse is the cuboid’s space diagonal."},{"text":"

Find the face/base diagonal by adding length and width.
Use that diagonal and the height in a Pythagoras calculation.

","correct":false,"explanation":"Adding length and width does not give the base diagonal; that mistake ignores the first right-angled triangle. Use Pythagoras for the base diagonal, then use Pythagoras again with the height."},{"text":"

Find the face/base perimeter using the side lengths.
Use half that perimeter and the height in Pythagoras.

","correct":false,"explanation":"Half the base perimeter is not a straight-line distance across the base. First calculate the base diagonal using Pythagoras, then combine it with the height using Pythagoras again."},{"text":"

Choose one base edge as the horizontal distance.
Use that edge and the height in a Pythagoras calculation.

","correct":false,"explanation":"One base edge omits the cuboid’s other horizontal dimension, so the result is only a face diagonal. Use the base diagonal with the height to obtain the space diagonal."}]},{"id":"b33983bb-c836-4990-9821-bb3e8a526e20","cardType":"basic","points":1,"front":"What is the shortcut formula for the space diagonal $d$ of a cuboid with length $l$, width $w$, and height $h$?","back":"$$d^2 = l^2 + w^2 + h^2$","mcqOptions":[{"text":"$$d^2 = l^2 + w^2 + h^2$","correct":true,"explanation":"Applying Pythagoras across the base and then through the cuboid gives the sum of all three squared perpendicular dimensions, so $d^2=l^2+w^2+h^2$."},{"text":"$$d^2 = l + w + h$","correct":false,"explanation":"This adds unsquared lengths, which is incompatible with Pythagoras and even gives mismatched units on the two sides. The correct formula is $d^2=l^2+w^2+h^2$."},{"text":"$$d = l^2 + w^2 + h^2$","correct":false,"explanation":"This forgets that the sum of the squares equals $d^2$, not $d$, producing a dimensional mismatch. The correct relationship is $d^2=l^2+w^2+h^2$."},{"text":"$$d^2 = (l + w + h)^2$","correct":false,"explanation":"Squaring the total introduces extra cross-terms such as $2lw$, which Pythagoras does not include. The three dimensions must be squared separately: $d^2=l^2+w^2+h^2$."}]},{"id":"1b6a409d-6077-4056-a0c2-46da3f03fb29","cardType":"cloze","points":null,"front":"To find a missing depth $x$ in a cuboid where the space diagonal $d$, width $w$, and height $h$ are known, the formula is $x^2 = \\text{[...]}$.","back":"To find a missing depth $x$ in a cuboid where the space diagonal $d$, width $w$, and height $h$ are known, the formula is $x^2 = \\boxed{d^2 - w^2 - h^2}$.","mcqOptions":[{"text":"$$d^2 - w^2 - h^2$","correct":true,"explanation":"Starting from $d^2=x^2+w^2+h^2$, subtracting both known squared dimensions isolates the missing depth: $x^2=d^2-w^2-h^2$."},{"text":"$$d^2 + w^2 + h^2$","correct":false,"explanation":"Adding the known squares again makes the missing depth larger than the space diagonal. Rearranging requires subtraction, giving $x^2=d^2-w^2-h^2$."},{"text":"$$d^2 - w^2 + h^2$","correct":false,"explanation":"This changes the sign of only one known dimension, so the height remains on the wrong side of the equation. Both known squares must be subtracted from $d^2$."},{"text":"$$d^2 - (w + h)^2$","correct":false,"explanation":"Adding width and height before squaring creates an unwanted $2wh$ term. Pythagoras uses separate squares, so the correct expression is $d^2-w^2-h^2$."}]},{"id":"87d75b75-32f9-4cd3-9b61-7ac126747fd6","cardType":"cloze","points":null,"front":"Common mistake: calculating $d = l + w + h$ for a cuboid is wrong because Pythagoras works with squares of lengths, not [ ].","back":"Common mistake: calculating $d = l + w + h$ for a cuboid is wrong because Pythagoras works with squares of lengths, not lengths added directly.","mcqOptions":[{"text":"lengths added directly","correct":true,"explanation":"Pythagoras adds squared perpendicular lengths, as in $d^2=l^2+w^2+h^2$. Simply adding the original lengths measures a route along edges rather than the straight diagonal."},{"text":"lengths multiplied directly","correct":false,"explanation":"Multiplication is not the specific mistake in $d=l+w+h$; that expression adds unsquared lengths. Pythagoras instead adds their squares, so the missing phrase is lengths added directly."},{"text":"lengths subtracted directly","correct":false,"explanation":"Subtraction is not being performed in $d=l+w+h$; the misconception is adding the three unsquared dimensions. The correct contrast is with lengths added directly."},{"text":"angles added directly","correct":false,"explanation":"The formula $d=l+w+h$ contains lengths rather than angles, so angle addition is unrelated. Its error is treating the diagonal as lengths added directly instead of using squared lengths."}]},{"id":"26ccd6d4-2e76-4d40-a51e-619e94000a74","cardType":"basic","points":1,"front":"When finding the angle between a cuboid's diagonal and its base, what must be used as the adjacent side?","back":"The base diagonal (not just one edge of the base).","mcqOptions":[{"text":"The base diagonal (not just one edge of the base).","correct":true,"explanation":"The space diagonal’s projection onto the horizontal base runs between opposite base corners, so the adjacent side of the relevant right-angled triangle is the base diagonal."},{"text":"The length edge (not the diagonal across the base).","correct":false,"explanation":"Using only the length ignores the base’s width and forms the wrong triangle. The adjacent side must be the base diagonal, which is the space diagonal’s horizontal projection."},{"text":"The vertical height (not a side across the base).","correct":false,"explanation":"For an angle measured from the base, the vertical height is the opposite side, not the adjacent side. The adjacent side is the diagonal across the base."},{"text":"The space diagonal (not a side lying on the base).","correct":false,"explanation":"The space diagonal is the hypotenuse of the right-angled triangle, so it cannot also be the adjacent side. The adjacent side is its projection, the base diagonal."}]},{"id":"1a262f96-e531-4d7d-aae8-7b098e029290","cardType":"cloze","points":null,"front":"The [ ] of a line onto a plane is like its shadow on that plane.","back":"The projection of a line onto a plane is like its shadow on that plane.","mcqOptions":[{"text":"projection","correct":true,"explanation":"A projection represents where a line would fall on a plane, like a shadow cast onto that surface. It is used to form the appropriate two-dimensional triangle."},{"text":"intersection","correct":false,"explanation":"An intersection is the point or set of points where objects meet, not the line’s shadow on a plane. The shadow-like representation is called a projection."},{"text":"inclination","correct":false,"explanation":"Inclination describes the angle a line makes with a reference line or plane. The shadow-like line formed on the plane is instead called its projection."},{"text":"reflection","correct":false,"explanation":"A reflection produces a mirror image across a line or plane rather than a shadow onto it. The line’s shadow-like image on the plane is its projection."}]},{"id":"892c9721-5ccf-428e-a4bd-f592b3c6c7c6","cardType":"cloze","points":null,"front":"The projection of a line onto a plane is like its [ ] on that plane.","back":"The projection of a line onto a plane is like its shadow on that plane.","mcqOptions":[{"text":"shadow","correct":true,"explanation":"A projection shows where the line would appear on the plane when mapped directly onto it, like the shadow cast by the line onto that surface."},{"text":"reflection","correct":false,"explanation":"A reflection is a mirror image produced across a line or plane. The correct term is shadow because a projection represents the line’s mapped position on the plane."},{"text":"rotation","correct":false,"explanation":"A rotation turns an object about a fixed point or axis. The correct term is shadow because a projection shows the line mapped onto the plane."},{"text":"intersection","correct":false,"explanation":"An intersection is where geometric objects meet, which may be only one point. The correct term is shadow because the whole line is projected onto the plane."}]},{"id":"c8cf2adf-4f40-456c-8f7b-7305a52e25e6","cardType":"basic","points":1,"front":"How is the angle between a line and a plane defined?","back":"The angle between the line and its projection on the plane.","mcqOptions":[{"text":"The angle between the line and its projection on the plane.","correct":true,"explanation":"The projection represents the line’s direction along the plane, so the angle between the original line and this projection is defined as the line-plane angle."},{"text":"The angle between the line and the normal to the plane.","correct":false,"explanation":"The angle with the normal is complementary to the angle with the plane. The required angle is between the line and its projection on the plane."},{"text":"The angle between the projection and a perpendicular to the line.","correct":false,"explanation":"Using an arbitrary perpendicular to the line does not define the line-plane angle. It is measured between the line and its projection on the plane."},{"text":"The angle between the plane and a vertical line through the plane.","correct":false,"explanation":"A vertical line is not necessarily related to the given line and may depend on the diagram’s orientation. The angle uses the line and its projection."}]},{"id":"cbc9874a-c667-43ee-a7be-048da2261069","cardType":"basic","points":1,"front":"How do you find the projection of a point onto a plane?","back":"Drop a perpendicular line from the point down to the plane.","mcqOptions":[{"text":"Drop a perpendicular line from the point down to the plane.","correct":true,"explanation":"The perpendicular gives the shortest route from the point to the plane. Where it meets the plane is the point’s orthogonal projection."},{"text":"Draw a parallel line from the point across to the plane.","correct":false,"explanation":"A line parallel to the plane will not meet it, so it cannot locate the projection. A perpendicular must be dropped from the point to the plane."},{"text":"Draw a horizontal line from the point across to the plane.","correct":false,"explanation":"Horizontal depends on the diagram’s orientation and does not guarantee a 90-degree angle with the plane. The projection is found using a perpendicular line."},{"text":"Join a diagonal line from the point to a corner of the plane.","correct":false,"explanation":"Joining the point to a chosen corner gives a sloping segment, not its projection. The correct construction drops a perpendicular line to the plane."}]},{"id":"bb4b80f8-7cf5-4fbf-a3c7-dad353993cc8","cardType":"cloze","points":null,"front":"In a symmetrical square-based pyramid, the vertical height meets the base at the [ ] of the square.","back":"In a symmetrical square-based pyramid, the vertical height meets the base at the centre of the square.","mcqOptions":[{"text":"centre","correct":true,"explanation":"Symmetry places the apex directly above the centre of the square base, so the perpendicular vertical height meets the base at that central point."},{"text":"corner","correct":false,"explanation":"A line from the apex to a corner is a sloping edge, not the vertical height. In a symmetrical pyramid, the vertical height meets the centre."},{"text":"perimeter","correct":false,"explanation":"The perimeter is the boundary of the square rather than the point directly below the apex. The vertical height meets the base at its centre."},{"text":"edge","correct":false,"explanation":"An edge lies on the boundary of the base, so meeting it would place the apex off-centre. Symmetry requires the height to meet the centre."}]},{"id":"d3726a65-b803-48eb-af3f-34460b80aed1","cardType":"basic","points":2,"front":"In a pyramid, what is the difference between vertical height and a sloping edge?","back":"

Vertical height goes straight from apex to base at 90 degrees.
Sloping edge connects the apex to a corner of the base.

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

Vertical height goes straight from apex to base at 90 degrees.
Sloping edge connects the apex to a corner of the base.

","correct":true,"explanation":"Vertical height is perpendicular to the base and measures the pyramid’s true height. A sloping edge instead joins the apex to a base vertex."},{"text":"

Vertical height connects the apex to a corner of the base.
Sloping edge goes straight from apex to base at 90 degrees.

","correct":false,"explanation":"This reverses the two definitions. The vertical height is perpendicular to the base, while a sloping edge joins the apex to a corner of the base."},{"text":"

Vertical height runs along the base from corner to corner.
Sloping edge connects two corners of the base.

","correct":false,"explanation":"Both descriptions refer to lines within the base rather than lines from the apex. The height is perpendicular to the base, while a sloping edge reaches a base corner."},{"text":"

Vertical height follows a triangular face down to its base.
Sloping edge connects the apex to the centre of the base.

","correct":false,"explanation":"This confuses face height, vertical height and sloping edge. Vertical height reaches the base perpendicularly, while a sloping edge joins the apex to a base corner."}]},{"id":"ed75936b-b4df-4100-b9fe-31995b924e04","cardType":"basic","points":1,"front":"In a square-based pyramid, what is the relationship between the distance from corner to centre ($AO$) and the base diagonal ($AC$)?","back":"$$AO = \\frac{1}{2} AC$","mcqOptions":[{"text":"$$AO = \\frac{1}{2} AC$","correct":true,"explanation":"The diagonals of a square bisect each other at the centre. Therefore, the distance from a corner to the centre is half the full diagonal."},{"text":"$$AO = AC$","correct":false,"explanation":"This treats the corner-to-centre distance as the whole diagonal. Since the centre bisects the diagonal, the correct relationship is $AO = \\frac{1}{2}AC$."},{"text":"$$AO = 2AC$","correct":false,"explanation":"This reverses and doubles the relationship, making a segment larger than the full diagonal containing it. In fact, $AO = \\frac{1}{2}AC$."},{"text":"$$AO = \\frac{1}{\\sqrt{2}} AC$","correct":false,"explanation":"The factor $\\frac{1}{\\sqrt{2}}$ relates a square’s side to its diagonal, not half a diagonal. Since $O$ is the midpoint, $AO = \\frac{1}{2}AC$."}]},{"id":"400e075a-79b6-48af-8a49-53b14c87374b","cardType":"basic","points":1,"front":"If a square base has side length $s$, what is the formula for the full base diagonal $d$?","back":"$$d = \\sqrt{s^2 + s^2}$ or $d = s\\sqrt{2}$","mcqOptions":[{"text":"$$d = \\sqrt{s^2 + s^2}$ or $d = s\\sqrt{2}$","correct":true,"explanation":"The diagonal is the hypotenuse of a right-angled triangle with two sides of length $s$. Pythagoras gives $d^2=s^2+s^2$, so $d=s\\sqrt{2}$."},{"text":"$$d = s + s$ or $d = 2s$","correct":false,"explanation":"Adding the side lengths calculates a two-side path around the square, not the direct diagonal. Pythagoras gives $d=\\sqrt{s^2+s^2}=s\\sqrt{2}$."},{"text":"$$d = \\sqrt{s^2 - s^2}$ or $d = 0$","correct":false,"explanation":"This subtracts the squares instead of adding them, giving an impossible zero diagonal. Pythagoras requires $d=\\sqrt{s^2+s^2}=s\\sqrt{2}$."},{"text":"$$d = \\frac{s}{\\sqrt{2}}$ or $d = \\frac{\\sqrt{2}}{2}s$","correct":false,"explanation":"This gives half the square’s diagonal, which is the corner-to-centre distance. The full diagonal is $d=\\sqrt{s^2+s^2}=s\\sqrt{2}$."}]},{"id":"4f09cc5d-4333-41aa-8e0a-7494471d0de0","cardType":"cloze","points":null,"front":"The volume of a pyramid is found using the formula $V = \\text{[...]} \\times \\text{base area} \\times \\text{vertical height}$.","back":"The volume of a pyramid is found using the formula $V = \\boxed{\\frac{1}{3}} \\times \\text{base area} \\times \\text{vertical height}$.","mcqOptions":[{"text":"$$\\frac{1}{3}$","correct":true,"explanation":"A pyramid with a given base area and perpendicular height has one-third the volume of the corresponding prism, so the required multiplier is $\\frac{1}{3}$."},{"text":"$$\\frac{1}{2}$","correct":false,"explanation":"The factor $\\frac{1}{2}$ is used for triangular area, not pyramid volume. A pyramid occupies one-third of the corresponding prism, so the multiplier is $\\frac{1}{3}$."},{"text":"$$1$","correct":false,"explanation":"Using a multiplier of $1$ calculates the volume of a prism with the same base and height. A pyramid has one-third that volume, requiring $\\frac{1}{3}$."},{"text":"$$\\frac{1}{4}$","correct":false,"explanation":"Dividing by four is not the pyramid volume relationship and usually comes from confusing the number of triangular faces with volume. The correct multiplier is $\\frac{1}{3}$."}]},{"id":"e98e22b1-2538-4b8e-b3ff-11e01145f765","cardType":"basic","points":1,"front":"What trigonometry ratio is typically used to find the height ($O$pposite) of a pyramid when given the angle and the base projection ($A$djacent)?","back":"Tangent ($\\tan\\theta = \\frac{\\text{opp}}{\\text{adj}}$)","mcqOptions":[{"text":"Tangent ($\\tan\\theta = \\frac{\\text{opp}}{\\text{adj}}$)","correct":true,"explanation":"Tangent directly relates the opposite and adjacent sides. The pyramid’s perpendicular height is opposite the angle, while the relevant base projection is adjacent to it."},{"text":"Sine ($\\sin\\theta = \\frac{\\text{opp}}{\\text{hyp}}$)","correct":false,"explanation":"Sine relates the opposite side to the hypotenuse, not to the adjacent side. With the height and base projection involved, tangent is required because $\\tan\\theta=\\frac{\\text{opp}}{\\text{adj}}$."},{"text":"Cosine ($\\cos\\theta = \\frac{\\text{adj}}{\\text{hyp}}$)","correct":false,"explanation":"Cosine relates the adjacent side to the hypotenuse, so it does not directly connect the given base projection to the required height. Tangent links opposite and adjacent."},{"text":"Cotangent ($\\cot\\theta = \\frac{\\text{adj}}{\\text{opp}}$)","correct":false,"explanation":"Cotangent uses the required sides in the reversed order, which can lead to solving the relationship incorrectly. The standard ratio here is tangent, $\\tan\\theta=\\frac{\\text{opp}}{\\text{adj}}$."}]},{"id":"4bec2427-880b-4143-9c5a-a9200ff10df2","cardType":"basic","points":1,"front":"State the formula for $\\cos\\theta$ in a right-angled triangle.","back":"$$\\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}$","mcqOptions":[{"text":"$$\\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}$","correct":true,"explanation":"Cosine is the ratio of the side adjacent to the chosen angle to the hypotenuse. This is the CAH part of the mnemonic SOHCAHTOA."},{"text":"$$\\cos\\theta = \\frac{\\text{hypotenuse}}{\\text{adjacent}}$","correct":false,"explanation":"Reversing the fraction gives the reciprocal of cosine, called secant. Cosine itself is adjacent divided by hypotenuse, following the CAH part of SOHCAHTOA."},{"text":"$$\\cos\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}$","correct":false,"explanation":"Using the opposite side confuses cosine with sine. Cosine compares the adjacent side with the hypotenuse, so the correct formula is $\\cos\\theta=\\frac{\\text{adjacent}}{\\text{hypotenuse}}$."},{"text":"$$\\cos\\theta = \\frac{\\text{adjacent}}{\\text{opposite}}$","correct":false,"explanation":"Dividing adjacent by opposite is the reciprocal of tangent, not cosine. Cosine must use the hypotenuse as the denominator: $\\cos\\theta=\\frac{\\text{adjacent}}{\\text{hypotenuse}}$."}]},{"id":"3de549ce-bbc2-44ba-bad2-7427496ca8d0","cardType":"basic","points":1,"front":"State the formula for $\\sin\\theta$ in a right-angled triangle.","back":"$$\\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}$","mcqOptions":[{"text":"$$\\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}$","correct":true,"explanation":"Sine compares the side opposite the chosen angle with the hypotenuse. This is the SOH part of the mnemonic SOHCAHTOA."},{"text":"$$\\sin\\theta = \\frac{\\text{hypotenuse}}{\\text{opposite}}$","correct":false,"explanation":"Reversing the fraction gives the reciprocal of sine, called cosecant. Sine itself is opposite divided by hypotenuse, following the SOH part of SOHCAHTOA."},{"text":"$$\\sin\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}$","correct":false,"explanation":"Using the adjacent side confuses sine with cosine. Sine compares the opposite side with the hypotenuse, so the correct formula is $\\sin\\theta=\\frac{\\text{opposite}}{\\text{hypotenuse}}$."},{"text":"$$\\sin\\theta = \\frac{\\text{opposite}}{\\text{adjacent}}$","correct":false,"explanation":"Opposite divided by adjacent is tangent, not sine. Sine uses the hypotenuse as its denominator, giving $\\sin\\theta=\\frac{\\text{opposite}}{\\text{hypotenuse}}$."}]},{"id":"32a1794a-31bf-477f-b750-8b439511e5bf","cardType":"cloze","points":null,"front":"To maintain accuracy, you should keep [ ] during working and only round the final answer.","back":"To maintain accuracy, you should keep extra decimal places during working and only round the final answer.","mcqOptions":[{"text":"extra decimal places","correct":true,"explanation":"Retaining extra decimal places prevents small rounding errors from accumulating across several calculations. Rounding should be postponed until the final answer is obtained."},{"text":"rounded intermediate values","correct":false,"explanation":"Keeping rounded intermediate values introduces premature rounding errors, which can affect the final result. Extra decimal places should be retained throughout the working instead."},{"text":"whole-number approximations","correct":false,"explanation":"Whole-number approximations discard too much precision during working and can substantially alter the final answer. Extra decimal places should be kept until the calculation is complete."},{"text":"fewer decimal places","correct":false,"explanation":"Using fewer decimal places increases the loss of precision at each stage. Extra decimal places should be retained during working, with rounding performed only at the end."}]},{"id":"3fc10c99-cd8b-4598-ad08-2234361e4b43","cardType":"cloze","points":null,"front":"To maintain accuracy, you should keep extra decimal places during working and only [ ].","back":"To maintain accuracy, you should keep extra decimal places during working and only round the final answer.","mcqOptions":[{"text":"round the final answer","correct":true,"explanation":"Rounding only the final answer preserves accuracy throughout the calculation while allowing the result to be presented to the precision requested by the question."},{"text":"round every intermediate answer","correct":false,"explanation":"Rounding every intermediate answer allows small errors to accumulate, particularly in multi-step 3D calculations. Keep extra decimal places during working and round only the final answer."},{"text":"truncate the final answer","correct":false,"explanation":"Truncation simply removes later digits rather than rounding according to the next digit, creating a biased result. The final answer should be rounded to the requested accuracy."},{"text":"round at each calculation step","correct":false,"explanation":"Rounding after each calculation step repeatedly reduces precision and may change the final result. Retain extra digits at every stage and round only the final answer."}]},{"id":"75a3e03e-fd6b-41c7-b112-69d99cefe245","cardType":"basic","points":1,"front":"What is the first step recommended when solving 3D geometry problems in an exam?","back":"Mark the right angle and identify the specific 2D triangle being used.","mcqOptions":[{"text":"Mark the right angle and identify the specific 2D triangle being used.","correct":true,"explanation":"Marking the right angle and isolating the relevant 2D cross-section reveals which sides form a right-angled triangle, allowing Pythagoras or trigonometry to be applied correctly."},{"text":"Choose a trigonometric ratio and substitute all the values shown in the diagram.","correct":false,"explanation":"Choosing a trigonometric ratio before identifying the triangle risks labelling the wrong sides as opposite, adjacent or hypotenuse. First mark the right angle and isolate the relevant 2D triangle."},{"text":"Calculate the longest edge and use it as the hypotenuse of the triangle.","correct":false,"explanation":"Calculating the longest visible edge first may use an irrelevant triangle and wastes working. The reliable first step is to mark the right angle and identify the required 2D triangle."},{"text":"Sketch the whole solid and treat its outer edges as a single triangle.","correct":false,"explanation":"Treating the whole solid as one triangle ignores that trigonometric methods apply to a particular 2D cross-section. First mark the right angle and identify that specific triangle."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"3d Pythagoras and Trigonometry","title":"3d Pythagoras and Trigonometry Flashcards","topbar":{"backHref":"/igcse/maths/edexcel/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"24fc549b-bad5-4f56-9152-e210036c95d9","walled":"$L39"}],"$L3a","$L3b"]