Question

In triangle ABC, an altitude is drawn from vertex C to the line containing AB. The length of this altitude is h and h=AB. Which of the following is true?I. ∆ABC could be a right triangle.II. Angle C cannot be a right angle.III. Angle C could be less than 45°.​

Answer 1

Ответ:

Объяснение:

1. Yes, in this case we'll get, for example, a triangle with right angle A and altitude will match AC, as the same line.

2. No, an altitude CH will divide AB for 2 segments AH and BH, and $CH=\sqrt{AH*BH}$

3. No, angle C could be 45 or more, we can see it in Pythagorean theoreme.

Answer 2

Let's analyze the given information and each of the statements:

Given: In triangle ABC, an altitude is drawn from vertex C to the line containing AB. The length of this altitude is h, and it is given that h = AB.

I. ∆ABC could be a right triangle: For a right triangle, one of the angles is 90 degrees. If ∆ABC is a right triangle, angle C would be 90 degrees since the altitude drawn from vertex C to the hypotenuse AB forms two right angles with the legs AC and BC. However, in this case, it's given that h = AB, which implies that the altitude (h) is also the hypotenuse (AB). In a right triangle, the hypotenuse is always longer than the other two sides, so h cannot be equal to AB. Therefore, ∆ABC cannot be a right triangle in this scenario.

II. Angle C cannot be a right angle: From the given information, h = AB. In a right triangle, the altitude drawn to the hypotenuse is also the geometric mean of the two segments it divides the hypotenuse into. In this case, the altitude drawn from vertex C would divide the hypotenuse AB into two equal segments, each of length h/2. But it's given that h = AB. This would mean that the two segments would have zero length, which is not possible. Therefore, angle C cannot be a right angle.

III. Angle C could be less than 45°: Since the altitude drawn from vertex C to the line containing AB is equal in length to AB (h = AB), this means that the triangle is isosceles, with AC = BC. In an isosceles triangle, the angles opposite to the equal sides are also equal. Therefore, angles A and B are equal. If angle C were also less than 45 degrees, the sum of angles A, B, and C would be less than 180 degrees, which contradicts the fact that the angles of a triangle sum to 180 degrees. Therefore, angle C cannot be less than 45 degrees.

In conclusion, none of the statements I, II, or III is true based on the given information that h = AB.