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Triangle Special Facts Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

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 Triangles Types of Triangles Right Triangle 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 Types of Triangles Right Triangle

Fig. 2-13 . Right angle

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.

Triangles Types of Triangles Right Triangle

Fig. 2-13 . Right angle

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 

Triangles Special Facts Perimeter Of Triangle

Fig. 2-14 . Perimeter and area of triangle.

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?

Triangles Special Facts Perimeter Of Triangle

Fig. 2-14 . Perimeter and area of 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.

 

Practice problems for these concepts can be found at: Triangle Practice Test.

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