The sides of a triangle are related as 3: 4: 5. The perimeter of a triangle, the vertices of which are the midpoints

The sides of a triangle are related as 3: 4: 5. The perimeter of a triangle, the vertices of which are the midpoints of the sides of this triangle, is 16 cm. Find the sides and area of this triangle.

The sides of the triangle ∆А1В1С1, the vertices of which are the midpoints of the sides of this triangle ∆ABS, are the midlines of this triangle. The midline of a triangle is half the length of the side that is parallel to it:

A1B1 = AB / 2;

B1C1 = BC / 2;

A1C1 = AC / 2.

Since the sides of the triangle ∆ABS are related as 3: 4: 5, then the sides of the triangle ∆А1В1С1 will also be related as 3: 4: 5. Since the perimeter of the triangle ∆А1В1С1 is 16 cm, we express it like this:

3x – A1B1 side;

4x – side В1С1;

5x – A1C1 side;

3x + 4x + 5x = 16;

12x = 16;

x = 16/12 ≈ 1.3333;

A1B1 = 3 * 1.3333 ≈ 3.9999 ≈ 4 cm;

B1C1 = 4 * 1.3333 = 5.3 cm;

A1C1 = 5 1.3333 = 6.7 cm.

To calculate the area, we use Heron’s formula:

S = √p (p-a) (p-b) (p-c);

p = (a + b + c) / 2;

p = 16/2 = 8 cm;

S = √8 · (8 – 4) · (8 – 5.3) · (8 – 6.7) = √8 · 4 · 2.7 · 1.3 = √112.32 ≈ 10.6 cm2.

Answer: the sides of the triangle ∆А1В1С1 are 4 cm, 5.3 cm, 6.7 cm, and its area is 10.6 cm2.