Question

АВ - бісектриса трикутника АВС, АВ = 6 см, ВС = 10 см. Менший з відрізків, на які бісектриса ВК ділить сторону АС, дорівнює 3 см. Знайдіть АС Срочно будь-ласка!!!​

Answer 1

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Answer 2

Bisector of Triangle ABC

To find the length of AC in triangle ABC, where AB = 6 cm, BC = 10 cm, and the bisector VK divides side AC into two segments, with one segment measuring 3 cm, we can use the Angle Bisector Theorem.

According to the Angle Bisector Theorem, the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides opposite the angle.

Let's denote the length of segment AK as x and the length of segment KC as y. We can set up the following equation based on the given information:

x / y = AB / BC

Substituting the given values, we have:

x / y = 6 / 10

Simplifying the equation, we get:

x / y = 3 / 5

Since we know that one segment measures 3 cm, we can set up another equation:

x + y = 3

Now we have a system of equations:

x / y = 3 / 5 x + y = 3

We can solve this system of equations to find the values of x and y, which will give us the lengths of segments AK and KC.

Let's solve the system of equations:

From the first equation, we can express x in terms of y:

x = (3 / 5) * y

Substituting this expression for x into the second equation, we have:

(3 / 5) * y + y = 3

Simplifying the equation:

(8 / 5) * y = 3

Multiplying both sides by 5/8:

y = (3 * 5) / 8

y = 15 / 8

y = 1.875 cm

Now, we can substitute the value of y back into the expression for x:

x = (3 / 5) * 1.875

x = 1.125 cm

Therefore, segment AK measures 1.125 cm and segment KC measures 1.875 cm.

To find the length of AC, we can add the lengths of segments AK and KC:

AC = AK + KC

AC = 1.125 cm + 1.875 cm

AC = 3 cm

Therefore, the length of AC is 3 cm.