Answer to Question #310508 in Geometry for عبدالرحمن

Question

Given two similar triangles, one with small measurements that can be accurately determined, and the other with large measurements, but at least one is known with accuracy, can the other two measurements be deduced? Explain and give an example.

The similarity of triangles gives rise to trigonometry.

How could we understand that the right triangles of trigonometry with a hypotenuse of measure 1 represent all possible right triangles? Ultimately, the similarity of triangles is the basis for proportions between sides of two triangles, and these proportions allow for the calculations of which we are speaking here. The similarity of triangles is the foundation of trigonometry.

Answer 1

Given two similar triangles, one with accurate measurement while the other one only one measurement is accurately provided it is possible to deduce the other two sides of the triangles.

Justification

Given that △IJH is the smaller triangle and △XYZ is the larger triangle.

Let YZ be the known side in the larger triangle that corresponds to JH in the smaller triangle.

Since the two triangles are similar, their interior angles are the same and the sides that correspond to each in the two triangles are proportional to each other. Therefore, the difference between the two triangles is the enlargement factor (k). The enlargement factor can be found by dividing the known measurement of the larger triangle by its corresponding side in the smaller triangle. i,e

$\displaystyle\Rightarrow{k=\frac{{{Y}{Z}}}{{{J}{H}}}}$

Other sides of the bigger triangle can be found by multiplying the length of the smaller triangle with the enlargement factor,k.

XY = k* IJ

XZ= k* IH

Another means can be through using the COSINE and SIN rule as shown below;

COSINE formula:

a² = b² + c² − 2bc cos(A)

SIN formula:

$\frac{a}{sin(A)}=\frac{b}{sin(B)} =\frac{c}{sin(C)}$

In the case of a right angle triangle with hypotenuse 1 (can be mm, cm, m, or km)

$\frac{c}{sin(C)}=\frac{1}{sin(90^0)} =\frac{a}{sin(\alpha )}=\frac{b}{sin(\beta )}=1\\ \\$

Therefore, the size of the sides of this rectangular angle depends on the included angle $\alpha$ and $\beta$ which are complementary.

Hence if $\alpha = 30^0$

a = sin (30⁰) = 0.5

b= sin(60⁰) = 0.8660254

prove by pythogtas theorem

a²+b² = c²

0.5² + (0.8660254)² = 1

Therefore if you are provided with two similar triangles it is possible to deduce the size of the other two sides by using the enlargement factor, cosine, or sine rule. [Answer]