Question

Гіпотенуза прямокутного трикутника дорівнює 26 см, а катети відносяться як 5:12.Знайдіть катети цього трикутника та радіуси вписаного і описаного колаПОЖАЛУЙСТА ПОМОГИТЕ НОВАЯ УЧИТЕЛЬНИЦА ВСЕ ОЧЕНЬ СЕРЬЕЗНО, А ДР МЕНЯ НЕ ДОХОДИТ ПОЖАЛУЙСТО БЫСТРЕЕЕ

Answer 1

Дано: АВС - прямокутний трикутник; ∠С=90°., АВ = 26см.
Знайти: $AC,\,\,BC,\,\,r,\,\,R.$
     розв'язання:
Нехай коефіцієнт пропорційності буде х (см), тоді катет ВС - 5х (см), а катет АС - 12 х (см). Гіпотенуза АВ = 26см. За теормею Піфагора складаємо рівняння
$AB^2=BC^2+AC^2 \\ 26^2=(5x)^2+(12x)^2 \\ 676=25x^2+144x^2 \\ 676=169x^2 \\ x^2=676:169 \\ x^2=4 \\ x=2$
Отже, коефіцієнт пропорційности х =2. Катети тоді будуть: BC=5x=5*2=10 см и AC = 12x=12*2=24 см.
Радіус описаного кола дорівнює половині гіпотенузи
$R= \frac{AB}{2} = \frac{26}{2} =13$
А радіус вписанного кола
$r= \frac{BC+AC-AB}{2} = \frac{10+24-26}{2} =4$

Відповідь: $BC = 10\,\,cm,\,\,\,AC=24\,\,cm,\,\,\,r=4\,\,cm,\,\,\,R=13\,\,cm.$

Answer 2

Finding the lengths of the legs of a right triangle

To find the lengths of the legs of a right triangle when the hypotenuse and the ratio of the legs are given, we can use the following steps:

1. Let's assume that the lengths of the legs are 5x and 12x, where x is a common factor. 2. According to the given information, the hypotenuse is 26 cm. 3. We can use the Pythagorean theorem to solve for x. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. - In this case, we have (5x)^2 + (12x)^2 = 26^2. - Simplifying the equation, we get 25x^2 + 144x^2 = 676. - Combining like terms, we have 169x^2 = 676. - Dividing both sides by 169, we get x^2 = 4. - Taking the square root of both sides, we find that x = 2. 4. Now that we know the value of x, we can find the lengths of the legs: - The first leg is 5x = 5 * 2 = 10 cm. - The second leg is 12x = 12 * 2 = 24 cm.

Therefore, the lengths of the legs of the right triangle are 10 cm and 24 cm.

Finding the radii of the inscribed and circumscribed circles

To find the radii of the inscribed and circumscribed circles of a right triangle, we can use the following formulas:

1. The radius of the inscribed circle (r) can be found using the formula: - r = (a + b - c) / 2, where a and b are the lengths of the legs of the right triangle, and c is the length of the hypotenuse. - Substituting the values, we have r = (10 + 24 - 26) / 2 = 8 cm.

2. The radius of the circumscribed circle (R) can be found using the formula: - R = c / 2, where c is the length of the hypotenuse. - Substituting the value, we have R = 26 / 2 = 13 cm.

Therefore, the radius of the inscribed circle is 8 cm, and the radius of the circumscribed circle is 13 cm.

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