Similar Triangles

Similar Triangles: Two triangles are said to be similar if:
(i) Their corresponding angles are equal, AND
(ii) Their corresponding sides are in the same ratio (proportional).

If triangle ABC is similar to triangle DEF, we write: triangle ABC ~ triangle DEF (the symbol ~ denotes similarity).

This means:
Angle A = Angle D, Angle B = Angle E, Angle C = Angle F
AND
AB/DE = BC/EF = CA/FD = k (where k is the scale factor)

Order of Vertices Matters: When writing triangle ABC ~ triangle DEF, the order indicates the correspondence: A corresponds to D, B corresponds to E, and C corresponds to F. Writing the similarity with incorrect vertex correspondence is wrong, even if the triangles are similar. For example, if Angle A = Angle E, Angle B = Angle D, Angle C = Angle F, then the correct notation is triangle ABC ~ triangle EDF, NOT triangle ABC ~ triangle DEF.

Scale Factor: The common ratio k = AB/DE = BC/EF = CA/FD is called the scale factor of the similarity. If k = 1, the triangles are congruent (same size). If k > 1, triangle ABC is larger. If k < 1, triangle ABC is smaller.

Distinction Between Similar and Congruent:
All congruent triangles are similar (with scale factor 1), but similar triangles are not necessarily congruent. Similarity preserves shape; congruence preserves both shape and size.

Basic Proportionality Theorem (BPT) / Thales' Theorem:
If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. That is, if DE is parallel to BC in triangle ABC (with D on AB and E on AC), then AD/DB = AE/EC. The converse is also true: if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Properties of Similar Triangles:

(i) Corresponding angles of similar triangles are equal.
(ii) Corresponding sides are proportional.
(iii) The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides: Area(ABC)/Area(DEF) = (AB/DE)^2 = (BC/EF)^2 = (CA/FD)^2.
(iv) The ratio of perimeters equals the ratio of corresponding sides.
(v) The ratio of corresponding altitudes, medians, and angle bisectors equals the ratio of corresponding sides.

Let us derive the AA (Angle-Angle) Similarity Criterion, which is the most fundamental similarity criterion.

Theorem (AA Criterion): If in two triangles, two pairs of corresponding angles are equal, then the two triangles are similar.

Given: In triangle ABC and triangle DEF, Angle A = Angle D and Angle B = Angle E.

To Prove: Triangle ABC ~ Triangle DEF, i.e., AB/DE = BC/EF = CA/FD.

Construction: On ray DE, mark a point P such that DP = AB. On ray DF, mark a point Q such that DQ = AC. Join PQ.

Proof:

Step 1: In triangle ABC and triangle DPQ:
AB = DP (by construction)
Angle A = Angle D (given)
AC = DQ (by construction)
Therefore, triangle ABC is congruent to triangle DPQ (by SAS congruence criterion).

Step 2: From the congruence in Step 1:
Angle B = Angle DPQ (corresponding parts of congruent triangles)
But Angle B = Angle E (given)
Therefore, Angle DPQ = Angle E ... (i)

Step 3: In triangle DEF, since Angle DPQ = Angle DEF [from (i)], and these are corresponding angles formed by the transversal DE cutting lines PQ and EF, the line PQ is parallel to EF.

Step 4: Since PQ is parallel to EF in triangle DEF (with P on DE and Q on DF), by the Basic Proportionality Theorem:
DP/DE = DQ/DF ... (ii)

Step 5: But DP = AB and DQ = AC (by construction). Substituting in (ii):
AB/DE = AC/DF ... (iii)

Step 6: Similarly, by taking a different construction (marking points on sides DE and EF), we can show:
AB/DE = BC/EF ... (iv)

Step 7: From (iii) and (iv):
AB/DE = BC/EF = CA/FD

This, combined with the equal angles (Angle A = Angle D, Angle B = Angle E, and hence Angle C = Angle F since angle sum = 180 degrees), proves that triangle ABC ~ triangle DEF. QED.

Derivation of the Area Ratio Theorem:

If triangle ABC ~ triangle DEF with AB/DE = BC/EF = CA/FD = k, then:

Draw altitudes AM from A to BC and DN from D to EF.

In triangles ABM and DEN:
Angle ABM = Angle DEN (since Angle B = Angle E)
Angle AMB = Angle DNE = 90 degrees
Therefore triangle ABM ~ triangle DEN (AA criterion)
So AM/DN = AB/DE = k

Area(ABC)/Area(DEF) = (1/2 * BC * AM) / (1/2 * EF * DN)
= (BC/EF) * (AM/DN)
= k * k = k^2

Therefore, the ratio of areas of similar triangles is the square of the ratio of corresponding sides.

Similar triangles manifest in several geometric configurations and problem types in Class 10.

1. Triangles Made Similar by Parallel Lines:
When a line is drawn parallel to one side of a triangle, it creates a smaller triangle similar to the original. In triangle ABC, if DE is parallel to BC (D on AB, E on AC), then triangle ADE ~ triangle ABC by the AA criterion (Angle ADE = Angle ABC as corresponding angles, and Angle A is common). This is the most fundamental configuration.

2. Triangles in an X-Configuration (Two Secants):
When two lines intersect inside or outside two parallel lines, the resulting triangles are similar. If lines AB and CD intersect at point P, and AB is parallel to CD (or if angle APD = angle BPC as vertically opposite angles), triangles can be proved similar using AA.

3. Right Triangle Altitude Similarity:
In a right triangle ABC with the right angle at B, if an altitude BD is drawn to the hypotenuse AC, three similar triangles are created: triangle ABC ~ triangle ADB ~ triangle BDC. This configuration is crucial for deriving the Pythagoras theorem using similarity.

4. Shadow and Height Problems:
When the sun casts shadows, a person and their shadow form a right triangle similar to a pole and its shadow (since the sun's rays are parallel, creating equal angles). If a person of height h1 casts a shadow of length s1, and a pole of height h2 casts a shadow of length s2 at the same time, then h1/h2 = s1/s2.

5. Map and Scale Model Problems:
Maps and scale models are practical applications of similar triangles. If a map has a scale of 1:50000, every 1 cm on the map represents 50000 cm (500 m) on the ground. The triangles formed by distances on the map are similar to the actual triangles on the ground.

6. Overlapping Similar Triangles:
In many geometric proofs, similar triangles share a common vertex or side. For instance, two triangles sharing a vertex with one side of each lying along the same line create a configuration where AA similarity can be established.

7. Triangles Similar by SSS Criterion:
When all three pairs of corresponding sides are in the same ratio, the triangles are similar. This is often used when the problem provides all six side lengths and asks to check similarity.

8. Triangles Similar by SAS Criterion:
When two sides are proportional and the included angle is equal, the triangles are similar. This criterion is particularly useful when angle information is limited to the included angle.

Similar triangles have extensive applications in real-world measurement, engineering, and science.

Indirect Height Measurement: Before modern instruments, the height of tall structures like pyramids, towers, and cliffs was measured using similar triangles. By measuring the shadow of a stick of known height and the shadow of the structure at the same time, the height could be calculated proportionally. This technique was famously used by the Greek mathematician Thales to measure the height of the Great Pyramid of Giza around 600 BCE.

Surveying and Mapmaking: Cartographers use the principles of similar triangles to create accurate maps. Distances between locations on a map are proportional to actual distances, and the triangulation method (using similar triangles formed by known baseline distances and measured angles) allows surveyors to calculate distances to inaccessible points.

Architecture and Construction: Scale models of buildings are similar figures to the actual structures. Architects use these models to test designs. The ratio of areas (proportional to the square of the scale factor) is used to estimate material costs, and the ratio of volumes (cube of the scale factor) helps estimate concrete and space requirements.

Optics and Photography: The lens equation in optics relies on similar triangles formed by light rays passing through a lens. The magnification of an image (ratio of image height to object height) equals the ratio of image distance to object distance, which is derived from similar triangle relationships.

Navigation and Astronomy: Sailors historically used similar triangles to estimate their distance from shore by measuring angles. Astronomers use the same principle (parallax) to calculate the distance to nearby stars by measuring the angle they appear to shift when viewed from different positions in Earth's orbit.

Computer Graphics: Rendering 3D scenes on a 2D screen uses perspective projection, which is fundamentally based on similar triangles. Objects farther from the viewer appear smaller in proportion to their distance, following the rules of similar triangles.