Question

Катет прямокутного трикутника 12 см, а гіпотенуза 13 см, знайдіть довжини проекцій та другий катет

Answer 1

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По теореме Пифагора:

Квадрат длины гипотенузы равен сумме квадратов длин катетов.

Формула:

a² + b² = c²

а-катет.

b-катет.

с-гипотенуза.

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а=12см

b=?

c=13см

b²=13²-12²=169-144=25; b=√25 = 5см

Ответ: 5см.

Answer 2

Finding the Length of the Projections and the Second Cathetus

To find the length of the projections and the second cathetus of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right 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, we are given the length of one cathetus (12 cm) and the length of the hypotenuse (13 cm). Let's denote the length of the second cathetus as 'x'.

Using the Pythagorean theorem, we can set up the following equation:

12^2 + x^2 = 13^2

Simplifying the equation, we have:

144 + x^2 = 169

To solve for 'x', we subtract 144 from both sides of the equation:

x^2 = 169 - 144

x^2 = 25

Taking the square root of both sides, we find:

x = 5

Therefore, the length of the second cathetus is 5 cm.

To find the length of the projections, we can use similar triangles. The projections are perpendicular to the catheti and are formed by dropping perpendiculars from the vertices of the right angle to the catheti.

Since the triangles formed by the projections are similar to the original right triangle, we can use the concept of similarity to find their lengths.

Let's denote the length of the projection on the first cathetus as 'p1' and the length of the projection on the second cathetus as 'p2'.

Using the concept of similarity, we can set up the following proportion:

p1 / 12 = p2 / 5

Cross-multiplying, we have:

5p1 = 12p2

Simplifying the equation, we find:

p1 = (12/5)p2

Therefore, the length of the first projection is (12/5)p2.

To find the length of the second projection, we can use the fact that the sum of the lengths of the projections is equal to the length of the hypotenuse.

Therefore, we have:

p1 + p2 = 13

Substituting the value of p1 from the previous equation, we get:

(12/5)p2 + p2 = 13

Simplifying the equation, we find:

17p2/5 = 13

To solve for 'p2', we multiply both sides of the equation by 5/17:

p2 = (5/17) * 13

Simplifying further, we find:

p2 = 3.82

Therefore, the length of the second projection is approximately 3.82 cm.

To summarize: - The length of the second cathetus is 5 cm. - The length of the first projection is (12/5)p2. - The length of the second projection is approximately 3.82 cm.

Please note that the values of the projections are approximate due to rounding.