Question

Решите задачу, составив систему уравнения. Один из катетов прямоугольного треугольника на 3 см больше другого. Найдите периметр треугольника, если его гипотенуза 15 см. помогите срочно прошу очень ​

Answer 1

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Answer 2

To solve this problem, let's start by assigning variables to the lengths of the two legs of the right triangle. Let's call the shorter leg x cm, and the longer leg (x + 3) cm.

We know that the hypotenuse of the triangle is 15 cm. Using the Pythagorean theorem, we can write the equation:

x^2 + (x + 3)^2 = 15^2

Simplifying this equation will give us a quadratic equation that we can solve to find the value of x. Once we have the value of x, we can calculate the lengths of the two legs and the perimeter of the triangle.

Let's solve the equation step by step:

Step 1: Expand the equation

x^2 + (x + 3)^2 = 225

Step 2: Simplify the equation

x^2 + (x^2 + 6x + 9) = 225

Step 3: Combine like terms

2x^2 + 6x + 9 = 225

Step 4: Move all terms to one side of the equation

2x^2 + 6x + 9 - 225 = 0

Step 5: Simplify the equation further

2x^2 + 6x - 216 = 0

Now, we have a quadratic equation in standard form. To solve it, we can either factor it or use the quadratic formula. Let's use the quadratic formula:

Step 6: Apply the quadratic formula

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 2, b = 6, and c = -216. Substituting these values into the quadratic formula, we get:

x = (-6 ± √(6^2 - 4 * 2 * -216)) / (2 * 2)

Simplifying further:

x = (-6 ± √(36 + 1728)) / 4 x = (-6 ± √1764) / 4 x = (-6 ± 42) / 4

Now, we have two possible values for x:

x1 = (-6 + 42) / 4 = 36 / 4 = 9 x2 = (-6 - 42) / 4 = -48 / 4 = -12

Since we are dealing with lengths, the value of x cannot be negative. Therefore, x = 9 cm.

Now, we can find the lengths of the two legs: Shorter leg = x = 9 cm Longer leg = x + 3 = 9 + 3 = 12 cm

Finally, we can calculate the perimeter of the triangle by adding the lengths of all three sides: Perimeter = x + (x + 3) + hypotenuse Perimeter = 9 + 12 + 15 Perimeter = 36 cm

Therefore, the perimeter of the triangle is 36 cm.

I hope this helps! Let me know if you have any further questions.