Question

В треугольнике авс вм- медиана и вн- высота. Известно, что ас= 96 и вс=вм. найдете ан

Answer 1

Т.к.ВМ медиана,то АМ=МС=96:2=48 .    ВМ=ВС, значит треугольник ВМС-  равнобедренный.ВН-высота и медиана, значит МН=НС=48:2=24   АН=АМ+МН=  48+24=72

Answer 2

Triangle with Medians and Altitudes

In the given triangle ABC, AV is a median and VN is an altitude. It is known that AV = 96 and VS = VM. We need to find the length of AN.

To solve this problem, we can use the properties of medians and altitudes in a triangle.

Properties of Medians:

1. A median of a triangle divides the opposite side into two equal segments. 2. The three medians of a triangle intersect at a point called the centroid, which divides each median into a 2:1 ratio.

Properties of Altitudes:

1. An altitude of a triangle is perpendicular to the base. 2. The three altitudes of a triangle intersect at a point called the orthocenter.

Since AV is a median, it divides the side BC into two equal segments. Let's denote the length of BN as x. Therefore, NC will also be x.

Since VN is an altitude, it is perpendicular to the base BC. This means that the right-angled triangles AVN and BNC are similar.

Using the property of medians, we know that AV divides BC into two equal segments. Therefore, BV = VC = x.

Now, we can use the similarity of triangles AVN and BNC to find the length of AN.

In triangle AVN, we have: - AV = 96 (given) - VN = x (altitude) - AN = ?

In triangle BNC, we have: - BN = x (median) - NC = x (median) - BC = 2x (sum of the two medians)

Using the similarity of triangles AVN and BNC, we can set up the following proportion:

AV/BN = VN/NC

Substituting the given values, we get:

96/x = x/(2x)

Simplifying the equation, we have:

96/x = 1/2

Cross-multiplying, we get:

2x = 96

Dividing both sides by 2, we find:

x = 48

Therefore, the length of BN (and NC) is 48.

Since AV is a median, it divides BC into two equal segments. Therefore, BV = VC = 48.

To find the length of AN, we can use the Pythagorean theorem in triangle AVN:

AN^2 = AV^2 - VN^2 AN^2 = 96^2 - 48^2 AN^2 = 9216 - 2304 AN^2 = 6912

Taking the square root of both sides, we find:

AN = √6912 AN ≈ 83.07

Therefore, the length of AN is approximately 83.07.

Summary: In triangle ABC, with AV as a median and VN as an altitude, if AV = 96 and VS = VM, then the length of AN is approximately 83.07.