Question

Угол ACB=90 градусов AB=25 CD=12 AE-?

Answer 1

CD⊥AD как касательная и радиус проведённый в точку касаний. AE=AD как радиусы одной окружности.

ΔACD ~ ΔCBD по двум углам (∠CAD=∠BCD и ∠ADC=90°=∠CDB), из подобия следует следующее отношение:

$\dfrac{AD}{CD} =\dfrac{CD}{DB}$

Откуда AD·DB = CD² = 144.

Пусть AD=x, тогда DB = AB-AD = 25-x.

x·(25-x) = 144;

x²-25x+144=0;

x(x-16)-9(x-16)=0;

(x-16)(x-9)=0 ⇒ x₁=16; x₂=9.

Ответ: 9 или 16.

Answer 2

To find the length of AE, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, triangle ABC is a right-angled triangle with the right angle at point C (angle ACB = 90 degrees). We have the following information:

AB = 25 (one of the sides) CD = 12 (another side)

Let AE be the third side that we need to find. Since triangle ABC is a right-angled triangle, we can apply the Pythagorean theorem:

AC^2 + BC^2 = AB^2

Now, we need to find the lengths of AC and BC. Since CD is given, and AD is perpendicular to BC (as it forms a right angle), we can use the Pythagorean theorem again for triangle ADC:

AC^2 = AD^2 + CD^2

Substitute the given values:

AC^2 = 25^2 + 12^2 AC^2 = 625 + 144 AC^2 = 769

Now, solve for AC:

AC = √769 ≈ 27.7 (rounded to one decimal place)

Next, find BC using the Pythagorean theorem for triangle BCD:

BC^2 = BD^2 + CD^2

Since BD is equal to AC (both are sides of rectangle ABCD), we can write:

BC^2 = AC^2 + CD^2 BC^2 = 769 + 144 BC^2 = 913

Now, solve for BC:

BC = √913 ≈ 30.2 (rounded to one decimal place)

Finally, we need to find AE using the Pythagorean theorem for triangle ABE:

AE^2 = AB^2 - BE^2

Since BE is equal to BC, we can write:

AE^2 = AB^2 - BC^2 AE^2 = 25^2 - 30.2^2 AE^2 = 625 - 913.64 AE^2 = -288.64

Since the result is negative, it seems there might be an error in the provided information or calculation. Please double-check the values of AB, CD, or any other relevant information to ensure accuracy.