Question

Серединний перпендикуляр сторони АВ трикутника АВС перетинає сторону ВС у точці К. Знайдіть сторону АС, якщо ВС = 12 см, а периметр трикутника АКС = 18 см.

Answer 1

Розв'язання: 
Нехай дано АВС, МК - серединний перпендикуляр до сторони АВ, 
т. М належить сторон!і ВС, ВС = 16 см, Р∆АМС = 26 см. Знайдемо сторону АС. 
Розглянемо АМК i BMK.
1) АК = KB (т. К - середина АВ);
2) ∟AКM = ∟BKM = 90° (МК ┴ АВ); 
3) MК - спільна. 
Отже, АМК = ∆BMК за I ознакою, з цього випливає, що AM = MB.
Р∆АМС = АС + АМ + СМ (т.я. АМ = МВ, то Р∆АМС = АС + МВ + СМ).
26 = АС + MB + CM, MB + СМ = СВ = 16 см. 
26 = АС + 16; АС = 26 - 16; АС = 10 см. 
Biдповідь: AC = 10 см.

Answer 2

Finding the length of side AC in triangle ABC

To find the length of side AC in triangle ABC, we can use the concept of the mid-segment or midline of a triangle. The mid-segment is a line segment that connects the midpoints of two sides of a triangle.

In this case, the mid-segment is the perpendicular bisector of side AB. Let's call the point where the mid-segment intersects side BC as point K.

We are given that side BC has a length of 12 cm and the perimeter of triangle AKS is 18 cm.

To find the length of side AC, we need to determine the length of side AK.

Solution:

1. Let's assume that the length of side AK is x cm.

2. Since the mid-segment is the perpendicular bisector of side AB, it divides side AB into two equal parts. Therefore, the length of side BK is also x cm.

3. The perimeter of triangle AKS is given as 18 cm. This means that the sum of the lengths of sides AK, KS, and SA is 18 cm.

4. Since we know that side BK has a length of x cm, we can express the length of side KS in terms of x as well. Let's denote the length of side KS as y cm.

5. Using the information above, we can write the equation for the perimeter of triangle AKS:

AK + KS + SA = 18

x + y + (x + y) = 18

2x + 2y = 18

x + y = 9

6. Now, let's consider triangle ABC. We know that the mid-segment is the perpendicular bisector of side AB, which means that it divides side AB into two equal parts. Therefore, the length of side AC is twice the length of side AK.

AC = 2 * AK

AC = 2 * x

7. We need to find the value of x to determine the length of side AC. To do this, we can solve the system of equations formed by the equations in steps 5 and 6.

x + y = 9 (Equation 1)

AC = 2 * x (Equation 2)

8. Solving the system of equations, we can substitute Equation 1 into Equation 2:

AC = 2 * (9 - y)

9. To find the value of y, we can use the fact that the mid-segment is the perpendicular bisector of side AB. This means that the length of side KS is equal to half the length of side BC.

KS = BC / 2

y = 12 / 2

y = 6

10. Substituting the value of y into Equation 2, we can find the length of side AC:

AC = 2 * (9 - 6)

AC = 2 * 3

AC = 6 cm

Therefore, the length of side AC in triangle ABC is 6 cm.

Please note that the given search results did not provide specific information about the mid-segment or the length of side AC in triangle ABC. The solution provided above is based on the general principles of triangle geometry and the concept of the mid-segment.