Question

В прямоугольном треугольнике проекции катетов на гипотенузу равны 1 и 2 найдите периметр треугольника

Answer 1

Обозначим  катеты  а  и  b ; гипотенуза  с; ∠(а;с) = α  ⇒  ∠(b;c) = 90°- α ;
   a = 1: cosα = 1/cosα   ;  b = 2: cos(90°- α)= 2/sinα
     a² + b² = c² = (1+2)² = 9
    1/cos²α + 4/sin²α = 9
    sin²α + 4cos²α = 9sin²α·cos²α
    1 - cos²α + 4cos²α = 9·(1- cos²)·cos²α
     9·(cos²α)² - 6·cos²α + 1 = 0
    ( 3cos²α - 1)² = 0 
     cosα = √3/3   ⇒   sinα = √[1 - (√3/3)²] = √6/3
    a = 1/(√3/3) = √3  
    b = 2/(√6/3) = √6 

p = a + b + c = 3 + √3 + √6


   

Answer 2

Finding the Perimeter of a Right-Angled Triangle

To find the perimeter of a right-angled triangle, we can use the Pythagorean theorem and the given projections of the legs onto the hypotenuse.

Given: Projection of the first leg on the hypotenuse = 1 Projection of the second leg on the hypotenuse = 2

We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Using the given projections, we can express the lengths of the legs in terms of the hypotenuse: Let the lengths of the legs be x and y, then: x = 1 * c y = 2 * c

Using the Pythagorean theorem: c^2 = x^2 + y^2 c^2 = (1c)^2 + (2c)^2 c^2 = c^2 + 4c^2 c^2 = 5c^2 c = sqrt(5) * c

So, the length of the hypotenuse is c = sqrt(5) * c.

Now, the perimeter of the triangle is the sum of the lengths of all its sides: Perimeter = x + y + c

Substituting the values of x, y, and c: Perimeter = 1c + 2c + sqrt(5)c Perimeter = (1 + 2 + sqrt(5))c

Calculating the Perimeter

To find the perimeter, we need to calculate the value of (1 + 2 + sqrt(5))c, where c is the length of the hypotenuse.

Let's calculate the value of (1 + 2 + sqrt(5))c: (1 + 2 + sqrt(5))c = (3 + sqrt(5))c

Therefore, the perimeter of the triangle is (3 + sqrt(5))c.

Final Answer

The perimeter of the right-angled triangle with projections of the legs onto the hypotenuse equal to 1 and 2 is (3 + sqrt(5))c.