Keep the ratio order and corresponding lengths together
If the length ratio of the first figure to the second is 3:4, the factor from the first to the second is 4/3. The two terms compare corresponding lengths with the same role. You cannot combine one figure's base and another's sloping side without establishing that they correspond.
Similarity scales corresponding heights and diagonals, as well as sides, by the same factor. Angles are preserved; their numerical values are not multiplied by the length factor. Reversing the comparison gives a factor of 3/4. Keep the direction of the comparison throughout the calculation.
Read a midpoint segment as a half-scale corresponding side
The first exercise joins the midpoints of two sides of a triangle. For the explanation, call their shared vertex A, the other vertices B and C, and the midpoints of AB and AC M and N. These are names introduced in this text, rather than labels claimed to appear in the exercise's diagram below. Halving both the horizontal and vertical displacements from A to B and from A to C reaches M and N.
The horizontal and vertical differences from M to N are each half the corresponding differences from B to C. Because both components shrink by the same factor, MN is parallel to BC and half as long. For example, half of C's horizontal coordinate minus half of B's horizontal coordinate equals half of their original horizontal difference. The vertical calculation is identical. This reasoning does not depend on a horizontally drawn base.
Triangle AMN is a half-scale copy of triangle ABC about A. The answer must state both the direction relationship, parallel, and the length relationship, half as long. A choice with only one correct part is insufficient. The presence of two midpoints is not a reason to double the segment's length.
Two length factors enter an area
Let the common length factor be k. A rectangle with dimensions p and q becomes one with dimensions kp and kq. Its new area, (kp) × (kq), is k² times the original pq. A triangle's base and perpendicular height likewise both scale by k, introducing two factors into base × height / 2.
A polygon can be divided into nonoverlapping triangles. If every triangular area scales by k², adding their areas gives the same factor for the whole polygon. For ordinary figures with curved boundaries, approximate their areas from inside and outside with small rectangles. Every rectangular area and the gap between the approximations scale by k². Making that gap smaller carries the same factor to the area of the whole figure.
The second exercise gives length ratio 3:4, so the area ratio is 3²:4² = 9:16. Cubing gives 27:64, the ratio used for volumes of similar solids. Leaving 3:4 unchanged retains only one length factor. Producing 6:16 by doubling one term but squaring the other also fails to compare the two areas by the same rule.
Three directional factors enter a volume
A cuboid with dimensions p, q and r has volume pqr. Scaling every direction by k gives (kp) × (kq) × (kr), or k³ times that volume. As an invented comparison, doubling a 2 cm by 3 cm by 4 cm cuboid gives dimensions 4 cm by 6 cm by 8 cm. Its volume changes from 24 cm³ to 192 cm³, a factor of eight rather than two.
For ordinary solids with curved surfaces, approximate the volume from inside and outside with small cuboids. Uniform scaling multiplies each cuboid's volume and the difference between the approximations by k³. As the approximation gap becomes smaller, the original and scaled solid's volumes retain that relationship. This is not a claim that a curved solid can be tiled exactly with finitely many cuboids.
The third exercise has length ratio 1:3, giving volume ratio 1³:3³ = 1:27. The ratio 1:9 squares the terms for area; 1:3 retains the original length ratio. The ratio 3:1 also reverses the comparison order. Notice that the midpoint, area and volume exercises supply different length ratios: they are separate comparisons.
Distinguish uniform scaling from a one-direction stretch or unit conversion
The k² and k³ rules require every corresponding length to change by the same factor. Change an invented 2 cm by 3 cm rectangle to 4 cm by 3 cm: only one direction doubles, while the other stays fixed. Its area doubles from 6 cm² to 12 cm², but these rectangles are not similar. Knowing that one side doubles does not establish a fourfold area increase.
Under uniform doubling, the 2 cm by 3 cm rectangle instead becomes 4 cm by 6 cm, with area 24 cm². Its perimeter is a sum of side lengths and doubles from 10 cm to 20 cm. Area and perimeter thus have different factors under the same enlargement. Reduction works in the same way: a half-scale copy has half the lengths, one quarter of the area and one eighth of the volume.
Unit conversion expresses the same quantity in another unit. Writing 2 cm as 20 mm does not enlarge the length itself. Here geometric enlargement or reduction changes the figure's lengths and areas measured in the same units. First distinguish a change of shape size from a change of measurement unit.
Reconstruct the exercises in the order length, area and volume
The exercises below ask about a midpoint segment, an area ratio of similar plane figures and a volume ratio of similar solids. Before opening an answer, write the requested measure and the direction of the comparison. After reading, hide the explanation and reconstruct the parallel-and-half relationship, a product of two lengths and a product of three lengths in your own words. Retain the original exercise and diagram conditions.
If matching a base to its height is difficult, return to the prerequisite “Geometry: read the conditions of the shape.” Review ratio direction in “Fractions and ratios: name the whole.” Continue by comparing the related “Changing units while keeping the quantity” with the geometric scaling here, which changes the quantity. “Expressions: give the unknown a role” helps you assign roles to letters, such as k for the factor and S for the original area in k²S. The practice preset selects middle-school-level, standard geometry questions, including matching questions beyond these three examples.