Question

В треугольнике ABC BM - медиана и BH - высота. Известно, что АС = 96 и BC = BH. Найдите AH.Напишите пожалуйста с подробным решением((

Answer 1

1) Рассмотри треугольник АВМ и треугольник ВМС у них
АМ=МС ( т. к ВМ медиана)
угол М = углу М ( ВН высота)
ВС=АВ (условие)
Из этого следует, что они равны.
2) Треугольник АВС равнобедренный так как маленькие треугольники равны и все части у них тоже равны ( углы при основании)
3) А в равнобедренном треугольники медиана является высотой и гипотенузой
Следовательно 96/2=48
Ответ:48

Answer 2

Given Information

We are given that in triangle ABC, BM is the median and BH is the height. It is also known that AC = 96 and BC = BH. We need to find the length of AH.

Solution

To find the length of AH, we can use the properties of medians and heights in a triangle.

Step 1: Let's start by drawing triangle ABC and labeling the given information.

         A
        / \
       /   \
      /     \
     /       \
    B---------C

We know that AC = 96 and BC = BH. Let's label these lengths on the triangle.

         A
        / \
       /   \
      /     \
     /       \
    B---------C
     \       /
      \     /
       \   /
        \ /
         M

Step 2: Since BM is the median, it divides side AC into two equal parts. Let's label the point where BM intersects AC as D.

         A
        / \
       /   \
      /     \
     /       \
    B---------C
     \       /
      \     /
       \   /
        \ /
         M
         |
         D

Since BM is the median, we have AD = DC.

Step 3: Now, let's consider the right-angled triangle BHD. BH is the height, and we know that BC = BH. This means that triangle BHD is an isosceles right-angled triangle.

         A
        / \
       /   \
      /     \
     /       \
    B---------C
     \       /
      \     /
       \   /
        \ /
         M
         |
         D
         |
         H

In an isosceles right-angled triangle, the length of the hypotenuse is equal to √2 times the length of each leg. Therefore, we can write:

BD = BH = BC

Step 4: Now, let's consider the right-angled triangle AHD. We need to find the length of AH.

         A
        / \
       /   \
      /     \
     /       \
    B---------C
     \       /
      \     /
       \   /
        \ /
         M
         |
         D
         |
         H
         |
         |
         A

In triangle AHD, we have AD = DC and BD = BH. Since BM is the median, it divides AC into two equal parts, so AD = DC = 48.

Using the Pythagorean theorem, we can write:

AH^2 = AD^2 + HD^2

Since AD = DC = 48, we have:

AH^2 = 48^2 + HD^2

Step 5: Now, let's consider triangle BHD again. Using the Pythagorean theorem, we can write:

BD^2 = BH^2 + HD^2

Since BD = BH = BC, we have:

BC^2 = BH^2 + HD^2

But we know that BC = BH, so we can substitute BC with BH:

BH^2 = BH^2 + HD^2

Simplifying, we get:

0 = HD^2

This means that HD = 0.

Step 6: Now, let's go back to the equation we obtained in Step 4:

AH^2 = 48^2 + HD^2

Since HD = 0, we have:

AH^2 = 48^2 + 0

Simplifying, we get:

AH^2 = 2304

Taking the square root of both sides, we find:

AH = √2304

Step 7: Finally, let's calculate the value of AH:

AH = √2304

AH = 48

Answer

Therefore, the length of AH is 48.

Please note that this solution assumes that the given information is accurate and that the triangle is a right-angled triangle.