Question

5. В прямокутному трикутнику AВС ( С = 90°) проведено висоту CD і медіану СЕ. Площі трикутників АСD і ЕСВ дорівнюють відповідно 4 см² і 10 см². Знайти AB.​

Answer 1

Ответ:

Объяснение:

Зная свойство медианы ∆ ( она делит его на два равно великих) имеем что $(СЕВ)=$(АЕС)=10

$(АЕС)=0,5*АЕ*СД=10. ,АЕ=АВ/2 подставим и выразим

СД= 10/ (0,5 АВ/2)= 40/ АВ. . [1].

Из ∆ АСД. $(АСД)=0,5АД*СД= 4

СД= 4/(0,5АД) = 8/АД. [2]

40/АВ = 8/АД

АД= АВ/5.

Из ∆CDE

S(CДE)=0,5*ДE*CD=6,.

CD=6/0,5ДE= 12/ДЕ. [3]

ДЕ=АВ/2 - АД. из предыдущего АД=АВ/5. => ДЕ=0,3АВ. [4]

,Используя св-во высоты прямоугольного треугольника,

СД^2 = АД*ЕД. [5]

Подставляем [3] и [4] получаем

(40/АВ)^2 =(АВ/5)*(0,3АВ)

1600/АВ^2 = АВ^2/0,06

АВ^4 = 1600*0,06=96=16*6

АВ= {4√96}=2*{4√6}

Два корня 4 степени их 6

Answer 2

Given Information:

We are given a right triangle ABC, where angle C is 90 degrees. In this triangle, a height CD and a median CE are drawn. The areas of triangles ACD and ECB are given as 4 cm² and 10 cm² respectively. We need to find the length of AB.

Solution:

To find the length of AB, we can use the relationship between the areas of triangles and their bases.

Let's denote the length of AB as x.

The area of triangle ACD is given as 4 cm². The base of triangle ACD is AC, which is equal to x. Therefore, we can write the formula for the area of triangle ACD as:

Area of ACD = (1/2) * AC * CD

Substituting the given values, we have:

4 = (1/2) * x * CD

Similarly, the area of triangle ECB is given as 10 cm². The base of triangle ECB is BC, which is equal to x. Therefore, we can write the formula for the area of triangle ECB as:

Area of ECB = (1/2) * BC * CE

Substituting the given values, we have:

10 = (1/2) * x * CE

Since CE is a median, it divides the base AB into two equal parts. Therefore, CE = 1/2 * AB.

Substituting this value into the equation for the area of ECB, we have:

10 = (1/2) * x * (1/2 * AB)

Simplifying this equation, we get:

10 = (1/4) * x * AB

Now, let's solve these two equations simultaneously to find the value of x and AB.

From the equation for the area of ACD, we have:

4 = (1/2) * x * CD

From the equation for the area of ECB, we have:

10 = (1/4) * x * AB

Since CD and AB are perpendicular to each other, we can use the Pythagorean theorem to relate them:

AB² = AC² + BC²

Since AC = x and BC = x, we have:

AB² = x² + x²

AB² = 2x²

Taking the square root of both sides, we get:

AB = √(2x²)

Now, let's substitute the value of AB in the equation for the area of ECB:

10 = (1/4) * x * √(2x²)

Simplifying this equation, we get:

40 = x * √(2x²)

Squaring both sides of the equation, we have:

1600 = 2x⁴

Dividing both sides by 2, we get:

800 = x⁴

Taking the fourth root of both sides, we have:

x = ∛800

Calculating the cube root of 800, we get:

x ≈ 9.2388

Therefore, the length of AB is approximately 9.2388 cm.

Answer:

The length of AB is approximately 9.2388 cm.