Acute triangles have several pairs of similar triangles after being made three sides high

After the ordinary (three sides are unequal) acute triangles are made three sides high, there are a total of 4 right triangles with each angle as an acute angle, and they are similar in pairs. Each group is to choose two out of 4, there are 4 * 3 2 1 = 6 pairs, there are three groups, a total of 18 pairs.

If it is an isosceles triangle, it means that two of the groups are the same, that is, choose two out of 8, and choose two out of 4: there are 8*7 2 1+4*3 2 1 = 34 pairs, and there is a group of obtuse triangles that correspond to congruence, a total of 35 groups.

If it is an equilateral triangle, that is, each group is the same, that is, choose two of the 12 right triangles, there are 12 * 11 2 1 = 66 pairs, three obtuse triangles are congruent, and choose two in it, that is, 3 * 2 2 1 = 3 pairs, a total of 66 + 3 = 69 pairs.

6 pairs. 1) The three high intersects at one point and is divided into 6 small right triangles, with equal angles to the apex, a total of 3 pairs of similar triangles.

2) The whole acute triangle has three angles, and from each angle, there are two right triangles that share this angle, a total of 3 pairs.

A total of 6 pairs.

I don't know how to define the metric "right", so I can only say that I counted 3 groups of triangles that are similar to each other, and each group has 4 similar triangles.

Answer: a. Analysis: Acute triangles.

The three high intersections of the triangle are at (one point), the three high intersections of the acute triangle are at the (inside) of the triangle, and the right triangle is an early beam.

The intersection of the three high lines is on the hypotenuse of the right triangle, while the intersection of the three high lines of the tonal triangle is on the outside of the triangle.

By corners. Judgment Method:

1. Acute triangle: The three inner angles of the triangle are less than 90 degrees.

2. Right triangle: one of the three internal deficit angles of the triangle is equal to 90 degrees, which can be recorded as RT.

3. Obtuse triangle.

One of the three interior angles of the triangle is greater than 90 degrees.

Sameness: The vertex is perpendicular to the straight line where the opposite edge is located, which is a perpendicular segment. Differences:

Right triangleThe middle two are right-angled sides and one is within a triangle; In an acute triangle, three of the heights are within the triangle; Obtuse trianglesThe middle two are high inside and outside the triangle, and one is inside the triangle.

The three high intersections of a triangle are at a point, which is called the vertical center of the triangle.

The vertical center of an acute triangle is on the inside of the triangle, the vertical center of a right triangle is the vertex of the right angle, and the vertical center of an obtuse triangle is on the outside of the triangle (intersected by the high extension lines).

Area formula for triangles:

where a and b are the two sides of the triangle, and c is the opposite angle of the side c).

Because the formula involves a sine value based on a right triangle, and the "sine" is discussed in a right triangle without the control of a circle, it was in the 16th century. Copernicus.

Austria's protégé.

The mathematician Rhaeticus (1514-1574), in his book The Trigonometry Criterion, described the sinusoidal function.

The definition of is directly based on the "right triangle", i.e. sin = opposite hypotenuse. Therefore, it can be concluded that it appeared after the 16th century.

The height of the triangle is a perpendicular line from one end of the triangle to its opposite side, and the line segment between the vertex of the triangle and its perpendicular foot on the opposite side is called the height of the triangle on this side. The triangle has three sides, so it has three heights.

It is obvious that an acute triangle has 3 strips high;

Right triangle

The right-angled edge is the height of the right-angled triangle, and the right-angled vertices are toward the hypotenuse.

The perpendicular line is the height of the hypotenuse;

Obtuse triangles

The perpendicular line of the pure angle vertex to the opposite edge is the height of the edge, and the perpendicular line of the extension line to the opposite edge is the height of the edge.

Therefore, it is not correct to say that a right triangle is only 1 bar high and an obtuse triangle is 2 lines high.

There is no doubt that an acute triangle is three highs, because one high from each of the three vertices of an acute triangle to its opposite side will naturally be three.

A right triangle doesn't have to be just one strip high. It can be two, because in a right triangle there are two right sides, and both right sides can be as high as the triangle.

There are three obtuse triangles. Make a line from the obtuse angle to its opposite side, which is a high, and two acute angles make an extension line on their opposite side, then the corresponding height is the perpendicular line from the acute angle to its opposite side of the extension line, which is its height. Therefore, the obtuse triangle will also have three heights.

The three heights of an acute triangle intersect at one point, and the intersection points are within the triangle. The intersection of the three high lines of a right triangle is on the hypotenuse of the right triangle, while the intersection of the three high lines of a ton triangle is on the outside of the triangle. The intersection of the three high lines of the triangle is called the vertical center of the triangle.

The vertical center of an acute triangle is within the triangle; The vertical center of the right triangle is on the right-angled vertex of the code scramble; The obtuse angle three-foot mold is hollow outside the horn shape of the three tombs. The vertical center is the intersection of three perpendicular lines made from each vertex of the triangle to its opposite edge. The triangle has three vertices, three vertical feet, and the seven vertical centers can give 6 groups of four-point contours.

There are 3 highs. From an acute triangle.

A pendulum of the sail vertices to the opposite side of it to make a perpendicular line.

The line segment between the perpendicular feet of the vertices is the height of the acute triangle, because the acute angle or hail triangle has 3 vertices. Therefore, the acute triangle has 3 heights, and the 3 heights of the acute triangle are all inside the triangle.

Make a perpendicular line from one vertex of an acute triangle to its opposite side, and the line segment between the perpendicular feet of the vertex is the height of the acute triangle, because an acute triangle has 3 vertices. Therefore, the acute triangle has 3 heights, and the 3 heights of the acute triangle are all within the triangle. In addition, the three heights of the right triangle coincide with the two right-angled sides, the third is inside the right triangle, the two acute vertices of the obtuse triangle to the opposite side are not the sum of the macro on the outside of the obtuse triangle, and the third is high on the inside of the obtuse triangle.