Triangles: Types and Uses

have you ever cut across a vacant lot we know it is shorter to Walk This Way than it is to walk around the corner one way to find out how much shorter is to count your steps but there is another more accurate way we'll find out about it as we learn about triangles as you probably know a triangle is a closed figure having three sides and three angles we can also Define a triangle this way a figure composed of three points and the line segments connecting them triangles are not only important to us in mathematics but are also commonly used as an important element in architecture and in engineering design we can identify triangles by relationships among their sides in an equilateral triangle the three sides are equal with two equal sides we can form an isoceles triangle a scalene triangle has no two equal sides the perimeter of any triangle is the sum of the lengths of the sides of the triangle triangles are identified not only by their sides but also by their angles since this triangle has one obtuse angle that is one angle which measures more than 90° it can also be called an obtuse triangle a right triangle is a triangle having one right angle an acute triangle is a triangle that has three acute angles that is the measure of each is is less than 90° the total number of degrees in the three angles of any triangle is 180° thus whenever we know the size of two angles we can always find the number of degrees in the third one with two triangles we can demonstrate congruency this is another important concept by indicating the line segments this way way we can see how triangle ABC fits exactly or coincides with triangle XYZ line AB coincides with line XY BC with YZ AC with XZ angle a coincides with angle x b with y and C with Z we see that if two triangles are congruent all six parts of one triangle three angles and three sides are equal respectively to the six parts of the other the triangles are also equal in area congruency will help us demonstrate how we derive this formula for the area of a triangle triangle area equals 12 the base B time the height H first on the base of the triangle we construct a rectangle the rectangle and the triangle both have the same base B and height H we see that triangle AFC is congruent to a d c we know that these two congruent triangles are equal in area in like manner triangle CBE is congruent to cdb we know that these two congruent triangles are equal in area thus triangle AFC and triangle CBE together are equal in area to the area of our original triangle ACB we know that the area of a rectangle equals the base time the height SO2 the base time the height must equal the area of a triangle if the original triangle has a base of 10 in and a height of 5 in the triangle has an area of 25 Square in one of the most important theorems about triangles was credited to the Greek mathematician Pythagoras it became known as the Pythagorean theorem the Pythagorean theorem applies to all right triangles the hypotenuse is the side opposite the right angle in the theorem the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides in this right triangle the sides are three units four units and and five units in length using the Pythagorean theorem let's see if the measure of this area plus the measure of this area is equal to the measure of the third area the square on the hypotenuse we proceed as follows 3^ 2ar is 9 4^ SAR is 16 9 + 16 = 25 the square on the hypotenuse using this Theory we can always find the third side if we know the other two remember the problem we saw earlier the Pythagorean theorem will help us find out how much distance this boy is saving we'll use as our unit of measure a section of sidewalk the path which is made by his Crossing is the hypotenuse C and we wish to find its length sides a b and c form a right triangle the length of side a is 20 sections of sidewalk and the length of side B is 15 sections of sidewalk using the Pythagorean theorem we can find the length of the unknown side C since we know the lengths of the other two sides our formula for calculating ating any one of these distances when the other two are known is a 2+ b^ 2 equal c^2 side a measures 20 units 20 squar is 400 side B measures 15 units 15 squared is 225 the sum of these two squares is 62 5 when we take the square root of 625 we find it is 25 25 sections of sidewalk is the length of side C the other two sides 20 and 15 total 35 subtracting 25 from 35 gives us 10 the boy has saved himself 10 sections of sidewalk in this problem we have a ladder 24 ft long the distance from the bottom of the ladder to the wall is 6 ft what then is the distance from the ground to the point where the ladder is resting against the wall in this case we want to find the length of side B using the formula a^ 2 + b^2 = c^2 we can make the following calculations side C representing the latter is 24 ft long 24 SAR is 576 side a the distance from the bottom of the ladder to the wall is 6 ft 6^ SAR is 36 subtracting 36 from 576 gives us 540 this is the square of side B to find the length of the side we take the Square t of 540 which is 23 and 210 side B the distance from the ground to the point where the ladder is resting against the wall is 23 and 210 ft how to find the length of an unknown side of a right triangle is just one of the important things we have learned about triangles we have also seen that triangle can be identified according to the measures of their sides or their angles we learn to find the area of a triangle by using the formula area equals 12 the base times the height as you continue with your study of mathematics you'll find that the solutions to many kinds of problems depend on your knowledge of triangles [Music]