Question

Докажите, что высота треугольника меньше полусуммы сторон, между которыми она заключена( Геометрия, 7кл) помогите пожалуйста))

Answer 1

Вот решение доказательство................................................

Answer 2

Proof that the height of a triangle is less than half the sum of the sides

To prove that the height of a triangle is less than half the sum of the sides, we can use the triangle inequality theorem. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's consider a triangle with sides a, b, and c, and let h be the height of the triangle with respect to side c. We want to prove that h < (a + b) / 2.

To begin the proof, we can assume without loss of generality that side c is the longest side of the triangle. This assumption does not affect the generality of the proof because we can always relabel the sides if necessary.

Now, let's consider the right triangle formed by dropping a perpendicular from the vertex opposite side c to side c. This perpendicular is the height of the triangle.

Using the Pythagorean theorem, we can express the length of side c in terms of the lengths of the other two sides, a and b:

c^2 = a^2 + h^2

Simplifying this equation, we get:

h^2 = c^2 - a^2

Since c is the longest side, c > a, so c^2 > a^2. Therefore, h^2 < c^2, which implies that h < c.

Now, let's consider the sum of the lengths of sides a and b:

a + b > c

Since h < c, we can substitute h for c in the above inequality:

a + b > h

Dividing both sides of the inequality by 2, we get:

(a + b) / 2 > h

Therefore, we have proved that the height of a triangle is less than half the sum of the sides.

Note: The proof provided above is a general proof for any triangle. If you need a specific proof for a particular type of triangle (e.g., equilateral, isosceles, or scalene), please let me know, and I can provide a more specific proof.