VinaOnline.net

Properties-of-triangles-2

Properties of triangles – Part 2

Các thuộc tính của tam giác – Phần 2

similar: Đồng dạng

congruent: Tương đẳng

median: Đường trung tuyến

vertex: Đỉnh (số ít), vertices (số nhiều)

midpoint: Điểm giữa.

opposite side: Cạnh đối diện

cross: Cắt nhau, giao nhau, đi qua

angle bisectors: Các góc phân giác

center: Tâm

inscribed: Nội tiếp

circumscribed: Ngoại tiếp

altitude: Độ cao, đường cao

perpendicular bisectors: Đường trung trực

inequality: Bất đẳng thức

BÀI VIẾT LIÊN QUAN:

Similar triangles

Two triangles are said to be similar (having the same shape) if their corresponding angles are equal. The sides of similar triangles are in the same proportion. The two triangles below are similar because they have the same corresponding angles.

Properties-of-triangles-2-Pict1
Properties-of-triangles-2-Pict2

a : d = b : e = c : f

Example:

Two triangles both have angles of 30°, 70°, and 80°. If the sides of the triangles are as indicated below, find the length of side x.

Properties-of-triangles-2-Pict3
Properties-of-triangles-2-Pict4
Solution

The two triangles are similar because they have the same corresponding angles.
The corresponding sides of similar triangles are in proportion, so x : 3 = 6 : 4. This can be rewritten as \frac{x}{3}=\frac{6}{4}. Multiplying both sides by 3 gives x=\frac{18}{4} , or x=4\tfrac{1}{2}.

Congruent triangles

Two triangles are congruent (identical in shape and size) if any one of the following conditions is met:

1. Each side of the first triangle equals the corresponding side of the second triangle.

2. Two sides of the first triangle equal the corresponding sides of the second triangle, and their included angles are equal. The included angle is formed by the two sides of the triangle.

3. Two angles of the first triangle equal the corresponding angles of the second triangle, and any pair of corresponding sides are equal.

Example:

Triangles ABC and DEF in the diagrams below are congruent if any one of the following conditions can be met:

Properties-of-triangles-2-Pict5
Properties-of-triangles-2-Pict6

1. The three sides are equal

(sss) = (sss)

Properties-of-triangles-2-Pict7
Properties-of-triangles-2-Pict8

2. Two sides and the included angle are equal

(sas) = (sas)

Properties-of-triangles-2-Pict9
Properties-of-triangles-2-Pict10

3. Two angles and any one side are equal

(aas) = (aas) or (asa) = (asa)

Properties-of-triangles-2-Pict11
Properties-of-triangles-2-Pict12

Example:

In the equilateral triangle below, line AD is perpendicular (forms a right angle) to side BC. If the length of BD is 5 feet, what is the length of DC?

Properties-of-triangles-2-Pict13
Solution

Since the large triangle is an equilateral triangle, each angle is 60°.

Therefore ∠B is 60° and ∠C is 60°.

Thus, ∠B = ∠C. ADB and ADC are both right angles and are equal.

Two angles of each triangle are equal to the corresponding two angles of the other triangle.

Side AD is shared by both triangles and side AB = side AC.

Thus, according to condition 3 “Two angles and any one side are equal”, the two triangles are congruent.

Then BD = DC and, since BD is 5 feet, DC is 5 feet.

Medians

The medians of a triangle are the lines drawn from each vertex to the midpoint of its opposite side. The medians of a triangle cross at a point that divides each median into two parts:

  • One part of one-third the length of the median.
  • The other part of two-thirds the length.
Properties-of-triangles-2-Pict14

Angle bisectors

The angle bisectors of a triangle are the lines that divide each angle of the triangle into two equal parts. These lines meet in a point that is the center of a circle inscribed in the triangle.

Properties-of-triangles-2-Pict15

Altitudes

The altitudes of the triangle are lines drawn from the vertices perpendicular to the opposite sides. The lengths of these lines are useful in calculating the area of the triangle, since the area of the triangle is 1/2(base)(height), and the height is identical to the altitude.

Properties-of-triangles-2-Pict16
Properties-of-triangles-2-Pict17

Perpendicular bisectors

The perpendicular bisectors of the triangle are the lines that bisect and are perpendicular to each of the three sides. The point where these lines meet is the center of the circumscribed circle.

Properties-of-triangles-2-Pict18

The sum of any two sides of a triangle is greater than the third side.
Example:

If the three sides of a triangle are 4, 2, and x, then what is known about the
value of x?

Solution

Since the sum of two sides of a triangle is always greater than the third side, then:

4 + 2 > x

4 + x > 2

2 + x> 4

These three inequalities can be rewritten as

6 > x

x > -2

x > 2

For x to be greater than -2 and 2, it must be greater than 2.
Thus, the values of x are 2 < x < 6.

Tải bài viết

Leave a Comment