Triangles

Similar figures have the same shape but not necessarily the same size, while congruent figures have both the same shape and size. For example, all circles are similar, but only circles with the same radius are congruent. This distinction is crucial in understanding geometric properties and theorems.

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. For instance, in triangle ABC, if DE is parallel to BC, then AD/DB = AE/EC. This theorem is foundational for proving other similarity criteria.

Two triangles are similar if their corresponding angles are equal (AAA criterion). For example, if in triangles ABC and DEF, angle A = angle D, angle B = angle E, and angle C = angle F, then ABC ~ DEF. This criterion is useful when side lengths are not known but angles are measurable.

The SSS similarity criterion states that if the corresponding sides of two triangles are in the same ratio, then the triangles are similar. For example, if AB/DE = BC/EF = CA/FD, then triangle ABC is similar to triangle DEF. This criterion is used when all three sides are known.

The SAS similarity criterion states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar. For example, if angle A = angle D and AB/DE = AC/DF, then triangle ABC ~ triangle DEF.

Similarity of triangles is used in real life for indirect measurements, such as determining the height of a mountain or the distance of a faraway object. For instance, by using shadow lengths and similar triangles, the height of a tower can be calculated without direct measurement.

Similarity means triangles have the same shape but sizes may vary, while congruence means both shape and size are identical. All congruent triangles are similar, but not all similar triangles are congruent. This distinction is vital for solving geometric problems.

To find the length of a shadow, set up a proportion using similar triangles formed by the object and its shadow. For example, if a 6m pole casts a 4m shadow, and a tower casts a 28m shadow, the height of the tower is (6/4)*28 = 42m.

Corresponding angles in similar triangles are angles that occupy the same relative position. For example, if triangle ABC ~ triangle DEF, then angle A corresponds to angle D, angle B to angle E, and angle C to angle F. These angles are equal in measure.

The AA similarity criterion is important because it allows us to prove two triangles are similar by showing just two pairs of corresponding angles are equal. This simplifies proofs and calculations, especially when side lengths are not provided.

The Pythagorean Theorem can be proven using similar triangles. For a right-angled triangle ABC with right angle at B, drawing an altitude BD to the hypotenuse AC creates two smaller triangles similar to ABC. Using similarity ratios, we derive AB² + BC² = AC².

The RHS similarity criterion states that if the hypotenuse and one side of a right-angled triangle are proportional to the hypotenuse and one side of another right-angled triangle, the triangles are similar. This is a special case of the SAS criterion for right triangles.

In problems involving medians and similarity, use the property that the median divides the triangle into two smaller triangles of equal area. If two triangles are similar, their corresponding medians are proportional to their corresponding sides.

The scale factor is the ratio of corresponding sides of similar triangles. It determines how much larger or smaller one triangle is compared to the other. For example, if the scale factor is 2, all sides of the larger triangle are twice those of the smaller triangle.

To prove similarity using the Basic Proportionality Theorem, show that a line parallel to one side divides the other two sides proportionally. For example, in triangle ABC, if DE is parallel to BC and AD/DB = AE/EC, then triangle ADE ~ triangle ABC by AA criterion.

Common mistakes include assuming triangles are similar without verifying all criteria, confusing similarity with congruence, and misapplying the scale factor. Always check both angles and sides or use the correct similarity criterion to avoid errors.

To find an unknown side, set up a proportion using corresponding sides of similar triangles. For example, if triangle ABC ~ triangle DEF and AB/DE = BC/EF, then BC = (AB/DE)*EF. Ensure the sides correspond correctly in the proportion.

The Angle Bisector Theorem states that the angle bisector divides the opposite side in the ratio of adjacent sides. This theorem is useful in solving problems involving similar triangles and proportions, especially when angles and side ratios are involved.

In coordinate geometry, similar triangles can be identified by comparing slopes (angles) and distances (side lengths). If two triangles have proportional sides and equal corresponding angles, they are similar. This is useful in graphing and solving geometric problems.

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. For example, if the sides are in ratio 1:2, the areas are in ratio 1:4. This property is useful in solving problems involving area and proportionality.

By drawing an altitude to the hypotenuse of a right-angled triangle, three similar triangles are formed. Using the properties of similar triangles, the relationships between the sides can be derived, leading to the Pythagorean Theorem: a² + b² = c².

Similarity is fundamental in trigonometry because trigonometric ratios (sine, cosine, tangent) are based on the ratios of sides of similar right-angled triangles. This allows for consistent calculations regardless of the triangle's size.

Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional. For example, all squares are similar because their angles are equal (90°) and sides are proportional, but rectangles are not necessarily similar.

The AA criterion requires two pairs of corresponding angles to be equal, which is sufficient for similarity since the third angle is automatically equal (sum of angles is 180°). AAA is essentially the same but explicitly states all three angles are equal.

A mnemonic for similarity criteria is 'AA-SAS-SSS': AA for two equal angles, SAS for proportional sides and included angle, and SSS for all sides proportional. Remembering these helps quickly identify similar triangles in problems.