How to find the perimeter of a triangle?
Often, mathematical tasks require in-depth analysis, the ability to search for solutions and the choice of the necessary statements, formulas. In such work it is easy to get confused. Yet there are problems whose solution boils down to the use of a single formula. These tasks include the question of how to find the perimeter of a triangle.
Consider the basic formulas for solving this problem in relation to different types of triangles.
- The basic rule for finding the perimeter of a triangle is the following statement: the perimeter of a triangle is equal to the sum of the lengths of all its sides. Formula P = a + b + c. Here a, b, c are the lengths of the sides of the triangle, P is its perimeter.
- There are particular cases of this formula. For example:
- if the problem poses the question of how to find the perimeter of a right-angled triangle, then you can use both a classical formula (see § 1) and a formula that requires less data:P = a + b + √ (a2+ b2). Here a, b are the lengths of the legs of a right triangle.It is easy to see that the third party (hypotenuse) is replaced by the expression from the Pythagorean theorem.
- the perimeter of an isosceles triangle is found by the formulaP = 2 * a + b. Here a is the length of the side of the triangle, b is the length of its base.
- To find the perimeter of an equilateral (or regular) triangle, we calculate the value of the expressionP = 3 * awhere a is the length of the side of the triangle.
- To solve problems involving similar triangles, it is useful to know the following statement: the ratio of perimeters equals the coefficient of similarity. It is convenient to use the formula
P (ΔABC) / P (ΔA1B1C1) = k, where ΔABC ΔA1B1C1and k is the similarity coefficient.
Dan ΔABC with sides 6, 8, and 10 and ΔA1B1C1with sides 9, 12. It is known that angle B is equal to angle B1. Find the perimeter of triangle A1B1C1.
- Let AB = 6, BC = 8, AC = 10; A1B1= 9; B1C1= 12 Note that AB / A1B1= BC / B1C1because 6/9 = 8/12 = 2/3. Moreover, by the condition B = B1. These angles are between AB, BC and A sides.1B1A b1C1respectively. Conclusion - on the 2nd sign of similarity of triangles, ΔABC ΔA1B1C1. The coefficient of similarity k = 2/3.
- We find the formula p. 1 P (ΔABC) = 6 + 8 + 10 = 24 (units). You can use the formula of paragraph 2a, since the Pythagorean theorem proves that ΔABC is rectangular.
- It follows from item 2d, P (ΔABC) / P (ΔA1B1C1) = 2/3. Therefore, P (ΔA1B1C1) = 3 * P (ΔABC) / 2 = 3 * 24/2 = 36 (units).
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