Question

Помогите пожалуйста!Непойму как решить!Биссектриса Ke прямоугольного треуголькника KTP (<P=90 градусов) делит катет в отношении 5:13.Найдите косинус угла KTP.

Answer 1

треугольник КТР, уголР=90, КЕ-биссектриспа, РЕ/ТЕ=5/13, КР/КТ=РЕ/ТЕ, КР/КТ=5/13, КТ=13КР/5, cosK=sinT=КР/КТ=КР/(13КР/5)=5/13, cosТ(КТР)=корень(1-sinT в квадрате)=корень(1-25/169)=12/13

Answer 2

Problem Analysis

We are given a right triangle KTP, where angle P is 90 degrees. The bisector KE divides the leg KT into two segments in the ratio of 5:13. We need to find the cosine of angle KTP.

Solution

To find the cosine of angle KTP, we can use the properties of right triangles and the definition of cosine.

Let's assume that the length of the shorter segment of KT is 5x and the length of the longer segment is 13x. Therefore, the total length of KT is 5x + 13x = 18x.

Now, we can use the Pythagorean theorem to find the length of the hypotenuse TP. 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 our case, we have: TP^2 = KT^2 + KP^2

Since angle P is 90 degrees, KP is equal to KT. Therefore, we can rewrite the equation as: TP^2 = KT^2 + KT^2 TP^2 = 2KT^2

Taking the square root of both sides, we get: TP = sqrt(2KT^2) TP = sqrt(2(18x)^2) TP = sqrt(2 * 324x^2) TP = sqrt(648x^2) TP = 18sqrt(2)x

Now, we can use the definition of cosine to find the cosine of angle KTP. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

In our case, the adjacent side is KP and the hypotenuse is TP. Therefore, the cosine of angle KTP is: cos(KTP) = KP / TP

Substituting the values we have: cos(KTP) = KT / TP cos(KTP) = KT / (18sqrt(2)x)

Since KT is equal to KP, we can rewrite the equation as: cos(KTP) = KP / (18sqrt(2)x) cos(KTP) = (5x) / (18sqrt(2)x)

Simplifying the expression, we get: cos(KTP) = 5 / (18sqrt(2))

Therefore, the cosine of angle KTP is 5 / (18sqrt(2)).

Note: The above solution assumes that the given ratio of 5:13 is for the lengths of the segments of KT. If the ratio is for the lengths of KP and KT, the solution would be different.