Question

Допоможіть будь ласка! Дан прямокутний трикутник MKN. КТ висота, проведена з вершини прямого кута. Катети трикутника рівні 5 і 12. Знайти MN, KT, MT, TN.(з оформленням будь ласка)​

Answer 1

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Пояснення:

на фото нижче

Answer 2

Problem Statement

We are given a right triangle MKN, where KT is the altitude drawn from the vertex of the right angle. The lengths of the triangle's legs are 5 and 12. We need to find the lengths of MN, KT, MT, and TN.

Solution

To solve this problem, we can use the Pythagorean theorem and the properties of similar triangles.

Let's start by finding the length of MN. MN is the hypotenuse of the right triangle MKN. We can use the Pythagorean theorem, which 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, MN^2 = MK^2 + KN^2. Since MK is one of the legs of the triangle and its length is given as 5, and KN is the other leg of the triangle and its length is given as 12, we can substitute these values into the equation:

MN^2 = 5^2 + 12^2 MN^2 = 25 + 144 MN^2 = 169 MN = √169 MN = 13

Therefore, the length of MN is 13.

Next, let's find the length of KT. KT is the altitude drawn from the vertex of the right angle, and it divides the triangle into two smaller triangles, MKT and KNT. Since KT is perpendicular to MK, we can conclude that MKT and KNT are similar triangles.

The ratio of the lengths of corresponding sides in similar triangles is equal. Therefore, we can set up the following proportion:

KT/MK = KN/TN

Substituting the given values, we have:

KT/5 = 12/TN

Cross-multiplying, we get:

KT * TN = 5 * 12 KT * TN = 60

Since we don't have enough information to directly solve for KT and TN individually, we can use the fact that KT is the altitude of the right triangle MKN. This means that the area of the triangle MKN is equal to the product of the lengths of the legs divided by 2.

The area of MKN = (MK * KN) / 2 The area of MKN = (5 * 12) / 2 The area of MKN = 30

We can also calculate the area of MKN using the lengths of KT and TN. The area of MKN = (KT * TN) / 2.

Setting these two equations equal to each other, we have:

(KT * TN) / 2 = 30

Simplifying, we get:

KT * TN = 60

This is the same equation we obtained earlier. Therefore, we can conclude that KT = 5 and TN = 12.

Therefore, the length of KT is 5 and the length of TN is 12.

Finally, let's find the lengths of MT and TN. Since KT is the altitude of the right triangle MKN, MT and TN are the legs of the right triangle MKT.

Using the Pythagorean theorem, we can find the lengths of MT and TN:

MT^2 = MK^2 - KT^2 MT^2 = 5^2 - 5^2 MT^2 = 25 - 25 MT^2 = 0 MT = √0 MT = 0

TN^2 = KN^2 - KT^2 TN^2 = 12^2 - 5^2 TN^2 = 144 - 25 TN^2 = 119 TN = √119

Therefore, the length of MT is 0 and the length of TN is approximately 10.92.

To summarize: - MN = 13 - KT = 5 - MT = 0 - TN ≈ 10.92

Please let me know if you need any further assistance!