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THE 345 TRIANGLE The triangle shown in figure 1914 has its sides in the ratio 3 to 4 to 5. Any triangle with its sides in this ratio is a right triangle. It is a common error to assume that a triangle is a 345 type because two sides are known to be in the ratio 3 to 4, or perhaps 4 to 5. Figure 1915 shows two examples of triangles which happen to have two of their sides in the stated ratio, but not the third side. This Figure 1913.Finding unknown parts in a 45°90° triangle. Figure 1914.A 345 triangle. Figure 1915.Triangles which may be mistaken for 345 triangles. can be because the triangle is not a right triangle, as in figure 1915 (A). On the other hand, even though the triangle is a right triangle its longest side may be the 4unit side, in which case the third side cannot be 5 units long. (See fig. 1915 (B).) It is interesting to note that the third side in figure 1915 (B) is This is a very unusual coincidence, in which one side of a right triangle is the square root of the sum of the other two sides. Related to the basic 345 triangle are all triangles whose sides are in the ratio 3 to 4 to 5 but are longer (proportionately) than these basic lengths. For example, the triangle pictured in figure 196 is a 34 5 triangle. Figure 1916.Triangle with sides which are multiples of 3, 4, and 5. The 345 triangle is very useful in calculations of distance. If the data can be adapted to fit a 345 configuration, no tables or calculation of square root (Pythagorean Theorem) are needed. EXAMPLE: An observer at the top of a 40foot vertical tower knows that the base of the tower is 30 feet from a target on the ground. How does he calculate his slant range (direct line of sight) from the target? SOLUTION: Figure 1917 shows that the desired length, AB, is the hypotenuse of a right triangle whose shorter sides are 30 feet and 40 feet long. Since these sides are in the ratio 3 to 4 and angle C is 90°) the triangle is a 345 triangle. Therefore, side AB represents the 5unit side of the triangle. The ratio 30 to 40 to 50 is equivalent to 345, and thus side AB is 50 units long. Practice problems. Without reference to tables or to the rule of Pythagoras, solve the following problems: 1. An observer is at the top of a 30foot vertical tower. Calculate his slant range from a target on the ground which is 40 feet from the base of the tower. Figure 1917.Solving problems with a 345 triangle. 2. A guy wire 15 feet long is stretched from the top of a pole to a point on the ground 9 feet from the base of the pole. Calculate the height of the pole. Answers: 1. 50 feet 
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