Метрични зависимости в правоъгълен триъгълник

1. В правоъгълен триъгълник, катета лежащ срещу ъгъл 30° е 5см, та другият катет е:

а) 3√3;

б) 5√3;

в) 5√5;

г) 3√5;

The given question is asking for the length of the other leg of a right triangle when one leg is 5 cm and the angle opposite that leg is 30 degrees. In a right triangle, the sides are related by the trigonometric ratios. In this case, we can use the sine ratio, which states that the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. Since the angle is 30 degrees and the side opposite is 5 cm, we can set up the equation sin(30°) = 5/h, where h is the length of the hypotenuse. Solving for h, we get h = 5/sin(30°) = 5/(1/2) = 10 cm. Now, using the Pythagorean theorem, we can find the length of the other leg: sqrt(10^2 - 5^2) = sqrt(100 - 25) = sqrt(75) = 5√3 cm. Therefore, the correct answer is б) 5√3.

Explanation

The given question is asking for the length of the other leg of a right triangle when one leg is 5 cm and the angle opposite that leg is 30 degrees. In a right triangle, the sides are related by the trigonometric ratios. In this case, we can use the sine ratio, which states that the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. Since the angle is 30 degrees and the side opposite is 5 cm, we can set up the equation sin(30°) = 5/h, where h is the length of the hypotenuse. Solving for h, we get h = 5/sin(30°) = 5/(1/2) = 10 cm. Now, using the Pythagorean theorem, we can find the length of the other leg: sqrt(10^2 - 5^2) = sqrt(100 - 25) = sqrt(75) = 5√3 cm. Therefore, the correct answer is б) 5√3.

2. Не е правоъгълен триъгълника със страни:

а)√5, √11, 4;

б) √7, 3, 5;

в) √5, 2√5, 5;

г)√13, 2√3,5;

The given triangle with sides √7, 3, 5 does not satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, if we square the lengths of the sides, we get 7, 9, and 25. However, 7 + 9 does not equal 25, so the triangle is not right-angled.

Explanation

The given triangle with sides √7, 3, 5 does not satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, if we square the lengths of the sides, we get 7, 9, and 25. However, 7 + 9 does not equal 25, so the triangle is not right-angled.

3. Радиусът на описаната около правоъгълен триъгълник с катети 5см и 12см, окръжност е:

а) 6см;

б) 6,5см;

в) 13см;

г) 15см;

The radius of the circumcircle of a right-angled triangle can be found using the formula R = (a + b - c) / 2, where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. In this case, a = 5 cm and b = 12 cm. The hypotenuse can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. So, c^2 = 5^2 + 12^2 = 25 + 144 = 169. Taking the square root of both sides, c = √169 = 13 cm. Plugging these values into the formula for the radius, R = (5 + 12 - 13) / 2 = 4 cm. Therefore, the correct answer is б) 6,5см.

Explanation

The radius of the circumcircle of a right-angled triangle can be found using the formula R = (a + b - c) / 2, where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. In this case, a = 5 cm and b = 12 cm. The hypotenuse can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. So, c^2 = 5^2 + 12^2 = 25 + 144 = 169. Taking the square root of both sides, c = √169 = 13 cm. Plugging these values into the formula for the radius, R = (5 + 12 - 13) / 2 = 4 cm. Therefore, the correct answer is б) 6,5см.

4. Единият от катетите на правоъгълен триъгълник, лежащ срещу ъгъл 30° е 5см,а другият катет на триъгълника е :

а) 3√3;

б) 5√3;

в) 5√5;

г) 3√5;

The question states that one of the legs of a right triangle, opposite the 30° angle, is 5 cm. In a right triangle, the sides are related by the Pythagorean theorem. Since the angle opposite the 30° angle is 90°, the other leg can be found using the formula sin(30°) = opposite/hypotenuse. Solving for the hypotenuse, we get hypotenuse = opposite/sin(30°) = 5/0.5 = 10 cm. Using the Pythagorean theorem, we can find the length of the other leg: leg^2 + opposite^2 = hypotenuse^2. Substituting the known values, we get leg^2 + 5^2 = 10^2. Simplifying, we get leg^2 + 25 = 100. Solving for leg, we get leg = √75 = 5√3 cm. Therefore, the correct answer is б) 5√3.

Explanation

The question states that one of the legs of a right triangle, opposite the 30° angle, is 5 cm. In a right triangle, the sides are related by the Pythagorean theorem. Since the angle opposite the 30° angle is 90°, the other leg can be found using the formula sin(30°) = opposite/hypotenuse. Solving for the hypotenuse, we get hypotenuse = opposite/sin(30°) = 5/0.5 = 10 cm. Using the Pythagorean theorem, we can find the length of the other leg: leg^2 + opposite^2 = hypotenuse^2. Substituting the known values, we get leg^2 + 5^2 = 10^2. Simplifying, we get leg^2 + 25 = 100. Solving for leg, we get leg = √75 = 5√3 cm. Therefore, the correct answer is б) 5√3.

5. Височината към основата на равнобедрен триъгълник с бедро 15 и основа 6 е:

а) 6√6;

б)√61;

в) 3;

г) 4;

The height to the base of an isosceles triangle with legs 15 and base 6 can be found using the Pythagorean theorem. The height, represented by h, can be calculated as the square root of the difference between the square of the legs and half the square of the base. In this case, h = √(15^2 - (6/2)^2) = √(225 - 9) = √216 = 6√6. Therefore, the correct answer is а) 6√6.

Explanation

The height to the base of an isosceles triangle with legs 15 and base 6 can be found using the Pythagorean theorem. The height, represented by h, can be calculated as the square root of the difference between the square of the legs and half the square of the base. In this case, h = √(15^2 - (6/2)^2) = √(225 - 9) = √216 = 6√6. Therefore, the correct answer is а) 6√6.

6. Едната страна на правоъгълник е с 2 см по-малка от другата. Диагоналът му е 10 см. Намерете периметъра и лицето на правоъгълника.

а) 6,8

б) 14. 48

в) 24,48

г) 12,24

The correct answer is в) 24,48. To find the perimeter of the rectangle, we need to know the lengths of its sides. Let's assume that one side of the rectangle is x cm. According to the given information, the other side will be (x+2) cm. We can use the Pythagorean theorem to find x. The diagonal of the rectangle is the hypotenuse of a right triangle formed by the sides of the rectangle. So, we have x^2 + (x+2)^2 = 10^2. Solving this equation, we find x = 4 cm. Therefore, the sides of the rectangle are 4 cm and 6 cm. The perimeter is 2(4+6) = 2(10) = 20 cm. The area is 4*6 = 24 cm^2. So, the correct answer is в) 24,48.

Explanation

The correct answer is в) 24,48. To find the perimeter of the rectangle, we need to know the lengths of its sides. Let's assume that one side of the rectangle is x cm. According to the given information, the other side will be (x+2) cm. We can use the Pythagorean theorem to find x. The diagonal of the rectangle is the hypotenuse of a right triangle formed by the sides of the rectangle. So, we have x^2 + (x+2)^2 = 10^2. Solving this equation, we find x = 4 cm. Therefore, the sides of the rectangle are 4 cm and 6 cm. The perimeter is 2(4+6) = 2(10) = 20 cm. The area is 4*6 = 24 cm^2. So, the correct answer is в) 24,48.

7. Периметъра на ромб е 104 см, а отношението на диагоналите му е 5:12. Намерете диагоналите на ромба.

а)4,10

б) 20, 48

в) 10,24

г) 10,48

The perimeter of a rhombus is equal to the sum of all four sides. If we let each side of the rhombus be represented by "s", then the perimeter would be 4s. In this case, the perimeter is given as 104 cm, so we can set up the equation 4s = 104. Solving for s, we find that each side of the rhombus is 26 cm.

The ratio of the diagonals of a rhombus is equal to the ratio of the lengths of the sides. In this case, the ratio is given as 5:12. Since we know that each side is 26 cm, we can set up the equation 5/12 = 26/x, where x represents the length of the shorter diagonal. Solving for x, we find that the shorter diagonal is 10 cm.

To find the longer diagonal, we can use the Pythagorean theorem. The diagonals of a rhombus form four right triangles, where the diagonals are the hypotenuses. The sides of these right triangles are half the lengths of the sides of the rhombus. Using the shorter diagonal as the hypotenuse and half the length of a side as one of the legs, we can set up the equation a^2 + b^2 = c^2, where a = 13 cm (half the length of a side) and c = 10 cm (the shorter diagonal). Solving for b, we find that the longer diagonal is 24 cm.

Therefore, the diagonals of the rhombus are 10 cm and 24 cm, which corresponds to answer choice б) 20, 48.

Explanation

The perimeter of a rhombus is equal to the sum of all four sides. If we let each side of the rhombus be represented by "s", then the perimeter would be 4s. In this case, the perimeter is given as 104 cm, so we can set up the equation 4s = 104. Solving for s, we find that each side of the rhombus is 26 cm.

The ratio of the diagonals of a rhombus is equal to the ratio of the lengths of the sides. In this case, the ratio is given as 5:12. Since we know that each side is 26 cm, we can set up the equation 5/12 = 26/x, where x represents the length of the shorter diagonal. Solving for x, we find that the shorter diagonal is 10 cm.

To find the longer diagonal, we can use the Pythagorean theorem. The diagonals of a rhombus form four right triangles, where the diagonals are the hypotenuses. The sides of these right triangles are half the lengths of the sides of the rhombus. Using the shorter diagonal as the hypotenuse and half the length of a side as one of the legs, we can set up the equation a^2 + b^2 = c^2, where a = 13 cm (half the length of a side) and c = 10 cm (the shorter diagonal). Solving for b, we find that the longer diagonal is 24 cm.

Therefore, the diagonals of the rhombus are 10 cm and 24 cm, which corresponds to answer choice б) 20, 48.

8. Височината към хипотенузата в правоъгълен триъгълник е 2 см и я разделя на отсечки, едната от които е с 2 см по-малка от другата. Хипотенузата е :

а) 6

б) 8

в) 4

г) 10

The question states that the height to the hypotenuse of a right-angled triangle is 2 cm and it is divided into two segments, one of which is 2 cm smaller than the other. The hypotenuse can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Since the height is given as 2 cm, one segment of the height is 2 cm and the other segment is 4 cm. Using the Pythagorean theorem, we can calculate the hypotenuse as the square root of (4^2 + 2^2), which is equal to 10. Therefore, the correct answer is г) 10.

Explanation

The question states that the height to the hypotenuse of a right-angled triangle is 2 cm and it is divided into two segments, one of which is 2 cm smaller than the other. The hypotenuse can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Since the height is given as 2 cm, one segment of the height is 2 cm and the other segment is 4 cm. Using the Pythagorean theorem, we can calculate the hypotenuse as the square root of (4^2 + 2^2), which is equal to 10. Therefore, the correct answer is г) 10.