**Introduction to Triangle Special Facts**

**Right Triangle**

Suppose we have a triangle Δ *PQR* with sides *S, T* , and *U* , having lengths *s, t* , and *u* , respectively. If one of the interior angles of this triangle measures 90° (π/2 rad), an angle that is also called a *right angle* , then the triangle is called a *right triangle* . In Fig. 2-13, a right triangle is shown in which *PRQ* is a right angle. The side opposite the right angle is the longest side, and is called the *hypotenuse* . In Fig. 2-13, this is the side of length *u* .

Triangles have some special properties. These characteristics have applications in many branches of science and engineering.

**A Triangle Determines A Unique Plane**

The vertex points of a specific triangle define one, and only one, Euclidean (that is, flat) geometric plane. A specific Euclidean plane can, however, contain infinitely many different triangles. This is intuitively obvious when you give it a little thought. Just try to imagine three points that don’t all lie in the same plane! Incidentally, this principle explains why a three-legged stool never wobbles. It is the reason why cameras and telescopes are commonly mounted on tripods (three-legged structures) rather than structures with four or more legs.

**Sum Of Angle Measures**

In any triangle, the sum of the measures of the interior angles is 180° (π rad). This holds true regardless of whether it is an acute, right, or obtuse triangle, as long as all the angles are measured in the plane defined by the three vertices of the triangle.

**Triangle Formulas - Pythagorean Theorem, Perimeter, and Interior Area**

**Theorem Of Pythagoras**

Suppose we have a right triangle defined by points *P, Q* , and *R* whose sides are *S, T* , and *U* having lengths *s, t* , and *u* , respectively. Let *u* be the hypotenuse (Fig. 2-13). Then the following equation is always true:

*s* ^{2} + *t* ^{2} = *u* ^{2}

The converse of this is also true: If there is a triangle whose sides have lengths *s, t* , and *u* , and the above equation is true, then that triangle is a right triangle.

**Perimeter Of Triangle**

Suppose we have a triangle defined by points *P, Q* , and *R* , and having sides *S, T* , and *U* of lengths *s, t* , and *u* , as shown in Fig. 2-14. Then the perimeter, *B* , of the triangle is given by the following formula:

*B* = *s* + *t* + *u*

**Interior Area Of Triangle**

Consider the same triangle as defined above; refer again to Fig. 2-14. Let *s* be the base length, and let *h* be the height, or the length of a perpendicular line segment between point *P* and side *S* . The interior area, *A* , can be found with this formula:

*A* = *sh* /2

**Triangle Special Facts Practice Problems**

**PROBLEM 1**

Suppose that Δ *PQR* in Fig. 2-14 has sides of lengths *s* = 10 meters, *t* = 7 meters, and *u* = 8 meters. What is the perimeter *B* of this triangle?

**SOLUTION 1**

Simply add up the lengths of the sides:

*B* = *s* + *t* + *u*

= (10 + 7 + 8) meters

= 25 meters

**PROBLEM 2**

Are there any triangles having sides of lengths 10 meters, 7 meters, and 8 meters, in that order proceeding clockwise, that are not directly congruent to Δ *PQR* as described in Problem 1?

**SOLUTION 2**

No. According to the side–side–side (SSS) principle, all triangles having sides of lengths 10 meters, 7 meters, and 8 meters, in this order as you proceed in the same rotational sense, are directly congruent.

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