Question

Бічна сторона і основа рівнобедреного трикутника відповідно дорівнюють 100 і 120 см. Обчисліть довжини відрізків, на які медіана, проведена до основи, ділить висоту, опущену на бічну сторону. Буду дуже вдячний за допомогу!

Answer 1

Відповідь:     Р = 48 см .

Пояснення:

У прямок. ΔАВС  ∠С = 90° ;  точки дотику М , N , K кола відповідно

до гіпотенузи АВ , катетів ВС  і  АС . АМ - МВ = 4 см ; АВ = 20 см .

Нехай МВ = х см , тоді  АМ = ( х + 4 ) см .

      Рівняння :   x + x + 4 = 20 ;

                           2x = 16 ;

                             x = 8 cм ;   х + 4 = 8 + 4 = 12 ( см ) .

 За власт. дотичних  АК = АМ = 12 см ; BM = BN = 8 cм .

  КС = NC = r - радіус впис. в ΔАВС кола .

  АС = 12 + r ;  BC = 8 + r .  За Т. Піфагора маємо :  АВ² = АС² + ВС² ;

   ( 12 + r )² + ( 8 + r )² = 20² . Розкривши дужки і спростивши , маємо :

   r² + 20r - 96 = 0 ;   D = 784 > 0 ;  r₁ = - 24 < 0 ;  r₂ = 4 ;  r = 4 cм .  

   АС = 12 + 4 = 16 ( см ) ;  BC = 8 + 4 = 12 ( см ) . Периметр тр - ника

   Р = АВ + АС + ВС = 20 + 16 + 12 = 48 ( см ) ; Р = 48 см .

           

Answer 2

To solve this problem, we can use the properties of a median in a triangle. A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. In a triangle, all three medians intersect at a point called the centroid.

Let's denote the lengths of the sides of the triangle as follows: - The length of the base is 120 cm. - The length of the other two sides (which are equal in a isosceles triangle) is 100 cm each.

Finding the Length of the Median

To find the length of the median, we can use the formula:

Length of the Median = (1/2) * sqrt(2 * (a^2 + b^2) - c^2)

Where: - a and b are the lengths of the two equal sides of the triangle. - c is the length of the base of the triangle.

In this case, a = b = 100 cm and c = 120 cm.

Plugging in the values, we get:

Length of the Median = (1/2) * sqrt(2 * (100^2 + 100^2) - 120^2)

Calculating this, we find that the length of the median is approximately 111.80 cm.

Finding the Length of the Height

To find the length of the height, we can use the formula:

Length of the Height = sqrt(a^2 - (c/2)^2)

Where: - a is the length of the median. - c is the length of the base of the triangle.

In this case, a = 111.80 cm and c = 120 cm.

Plugging in the values, we get:

Length of the Height = sqrt(111.80^2 - (120/2)^2)

Calculating this, we find that the length of the height is approximately 92.13 cm.

Finding the Lengths of the Segments

To find the lengths of the segments, we need to divide the height into two equal parts. Since the median divides the height into two equal parts, we can use the following formula:

Length of the Segments = (1/2) * Length of the Height

Plugging in the value of the height, we get:

Length of the Segments = (1/2) * 92.13 cm

Calculating this, we find that the length of each segment is approximately 46.07 cm.

Therefore, the lengths of the segments into which the median divides the height, dropped from the vertex to the base, are approximately 46.07 cm each.

I hope this helps! Let me know if you have any further questions.