Question

Найдите отношение площадей треугольников АВС и KMN, если АВ = 8 см, ВС = 12 см, АС = 16 см, КМ = 10 см, MN = 15 см, NK = 20 см.Уважительная просьба с рисунком

Answer 1

Треугольники подобны, поэтому АB/KM=BC/MN=AC/NK

к=8/10=12/15=16/20=0,8 коэффициент подобия. Отношение площадей подобных треугольников равен коэффициенту подобия.  

Sabc/Skmn=0,64

По поводу чертежа - нужно нарисовать два треугольника, один из которых чуть меньше другого.

Надеюсь, что верно.

Answer 2

$S= \sqrt{p(p-a)(p-b)(p-c)}, p= \frac{a+b+c}{2}$ формула Герона
$p1= \frac{8+12+16}{2}=18;p2= \frac{10+15+20}{2}=22,5$
$S1= \sqrt{18*10*6*2} =12 \sqrt{15}$
$S2= \sqrt{22,5*12,5*7,5*2,5} = \frac{75}{4} \sqrt{15}$
$\frac{S1}{S2}= \frac{12 \sqrt{15} }{ \frac{75}{4} \sqrt{15} } = \frac{16}{25}$

Answer 3

Calculation of Triangle Areas

To find the ratio of the areas of triangles ABC and KMN, we need to calculate the areas of both triangles.

Triangle ABC has side lengths AB = 8 cm, BC = 12 cm, and AC = 16 cm. Triangle KMN has side lengths KM = 10 cm, MN = 15 cm, and NK = 20 cm.

To calculate the area of a triangle, we can use Heron's formula, which states that the area (A) of a triangle with side lengths a, b, and c is given by:

A = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle, given by:

s = (a + b + c) / 2

Let's calculate the areas of triangles ABC and KMN:

For triangle ABC: - AB = 8 cm - BC = 12 cm - AC = 16 cm

The semi-perimeter of triangle ABC is: s_ABC = (AB + BC + AC) / 2 = (8 + 12 + 16) / 2 = 36 / 2 = 18 cm

Using Heron's formula, the area of triangle ABC is: A_ABC = sqrt(s_ABC * (s_ABC - AB) * (s_ABC - BC) * (s_ABC - AC))

For triangle KMN: - KM = 10 cm - MN = 15 cm - NK = 20 cm

The semi-perimeter of triangle KMN is: s_KMN = (KM + MN + NK) / 2 = (10 + 15 + 20) / 2 = 45 / 2 = 22.5 cm

Using Heron's formula, the area of triangle KMN is: A_KMN = sqrt(s_KMN * (s_KMN - KM) * (s_KMN - MN) * (s_KMN - NK))

Now, let's calculate the areas:

For triangle ABC: - s_ABC = 18 cm - A_ABC = sqrt(18 * (18 - 8) * (18 - 12) * (18 - 16)) = sqrt(18 * 10 * 6 * 2) = sqrt(2160) ≈ 46.57 cm^2

For triangle KMN: - s_KMN = 22.5 cm - A_KMN = sqrt(22.5 * (22.5 - 10) * (22.5 - 15) * (22.5 - 20)) = sqrt(22.5 * 12.5 * 7.5 * 2.5) = sqrt(4218.75) ≈ 64.99 cm^2

Ratio of Triangle Areas

The ratio of the areas of triangles ABC and KMN can be calculated by dividing the area of triangle ABC by the area of triangle KMN:

Ratio = A_ABC / A_KMN

Substituting the calculated values:

Ratio = 46.57 cm^2 / 64.99 cm^2 ≈ 0.716

Therefore, the ratio of the areas of triangles ABC and KMN is approximately 0.716.

Please note that the calculations provided are based on the given side lengths of the triangles and the application of Heron's formula.