Use corresponding sides and angles in similar figures

Learning goal: Identify corresponding parts in similar triangles and special quadrilaterals, represent proportional relationships within and between them, and solve missing-length problems. Explain why one matching ratio alone is insufficient.

Similar figures have the same shape: corresponding angles match and corresponding lengths share a common positive scale factor. The figures may be in different positions or orientations. Begin by matching vertices and sides according to their roles, not by assuming that the leftmost side in one drawing corresponds to the leftmost side in another. Rotation or reflection changes appearance on the page without changing shape.

Consider a right triangle with side lengths three, four and five units and a second with corresponding lengths six, eight and ten. The right-angle vertices correspond, the shortest legs correspond and the longest sides correspond. Each large side divided by its corresponding small side is two: 6/3 = 8/4 = 10/5 = 2. The scale factor from the small triangle to the large is two; in the reverse direction it is one half.

The diagram preserves the same linear scale for both triangles, so every corresponding drawn side in the larger is twice as long. The area, however, is four times as large. The smaller right triangle has area (1/2)(3)(4) = six square units; the larger has area (1/2)(6)(8) = 24. Doubling both perpendicular dimensions multiplies area by two twice. Do not use the length factor as the area factor.

Proportionality can be expressed within a figure as well as between figures. The small triangle's leg ratio is 3/4 and the large triangle's matching leg ratio is 6/8, also 3/4. The hypotenuse-to-short-leg ratio is 5/3 in the first and 10/6 in the second. Maintain corresponding order: compare short leg to long leg in both triangles, 3/4 = 6/8. Comparing 3/4 with 8/6 mixes two orders and wrongly suggests the figures disagree. Taking the reciprocal of both ratios in the true equation keeps a single order (long leg to short leg) and gives 4/3 = 8/6.

Other triangles follow the same idea. An isosceles triangle with sides five, five and six can be enlarged by a factor of 3/2 to sides 7.5, 7.5 and nine. The equal-side relationship remains, and the base corresponds to the base. These are not right triangles, so do not automatically treat the given equal side as a perpendicular height when computing area. Similarity scales lengths; it does not turn every side into an altitude.

For rectangles, equal angles alone do not guarantee similarity because all rectangles have four right angles. A two-by-three rectangle is similar to a four-by-six rectangle: both dimensions double. A two-by-three rectangle is not similar to a four-by-five rectangle, because 4/2 = 2 while 5/3 is not two. The shared right angles must be accompanied by the corresponding side proportion.

Squares are similar to one another because all have equal angles and all four sides scale together. Rhombi need more attention: all sides in a rhombus are equal, but a square-shaped rhombus and a non-square rhombus have different angles and are not similar. A side-length list alone does not fix the shape of every quadrilateral. The construction lesson investigates that limitation directly.

To solve a missing length, establish a valid correspondence and constant factor. If triangles are given as similar and a corresponding side changes from four to ten, the scale factor from the smaller figure to the larger is 10/4 = 5/2. A side of six then becomes fifteen. If a large side is twenty, its small counterpart is twenty divided by 5/2, or eight. Multiplication and division depend on the direction of the comparison.

A proportion can express the same reasoning: x/6 = 10/4. Multiplying both sides by six gives x = 15. The equation is meaningful because x and six are corresponding lengths and the numerator/denominator orientation matches the known pair. Cross multiplication without a correspondence can produce a correct calculation for the wrong relationship.

Check all supplied information. A claim that sides three, four, five correspond to six, eight, eleven fails the common-factor test: the first two scale by two, but the third does not. If a drawing merely looks similar, that appearance is not enough to override inconsistent dimensions. State whether similarity is given, constructed or justified by appropriate conditions.

Worked correspondence: Triangle ABC is similar to DEF with A corresponding to D, B to E and C to F. If AB = 4, DE = 10 and BC = 6, then EF = 15. The order of the similarity statement tells which endpoints match. If the second triangle is rotated, that correspondence still holds.

Worked rectangle: A 3.5-by-five rectangle is enlarged by a factor of two. The new dimensions are seven and ten, while area grows from 17.5 to 70 square units. The length ratio is two and the area ratio is four. Both results follow from scaling the two perpendicular dimensions.

Worked mismatch: A student pairs the three-unit leg with an eight-unit leg and the four-unit leg with a six-unit leg merely because of page position. The ratios become 8/3 and 6/4, which disagree. Reidentify corresponding roles before concluding the figures are not similar.

Drawing task: Draw the three-four-five right triangle from perpendicular legs using a ruler, then enlarge each leg by a factor of two and connect the endpoints. Measure corresponding sides and angles within reasonable drawing precision. Label the exact intended dimensions separately from small measurement discrepancies. Rotate one drawing and explain why the correspondence remains.

Independent performance: Create and solve four missing-length problems involving a right triangle, a non-right triangle, a rectangle and a square. State the correspondence and scale-factor direction before calculating. Add one quadrilateral pair with proportional-looking sides but different angles, explaining why similarity is not established. Compare perimeter and area changes in one valid enlargement and justify why their factors differ.