How to find the median of a right triangle

Determining the median of a right triangle is one of the basic tasks in geometry. Often, its finding acts as an auxiliary element in the solution of some more complex task. Depending on the available data, the task can be solved in several ways.

You will need

geometry textbook.

Instruction manual

one

It is worth recalling that a triangle is rectangular if one and its angles is 90 degrees. And the median is a segment lowered from the corner of the triangle to the opposite side. Moreover, he divides it into two equal parts. In a right-angled triangle ABC, in which the ABC angle is right, the median BD, pubescent from the vertex of the right angle, equals half the hypotenuse AC. That is, in order to find the median, divide the hypotenuse value into two: BD = AC / 2. Example: Suppose that in a right triangle ABC (ABC-right angle), the values ​​of legs AB = 3 cm, BC = 4 cm are known., find the length of the median BD dropped from the vertex of the right angle. Decision:

1) Find the value of hypotenuse. By the Pythagorean theorem, AC ^ 2 = AB ^ 2 + BC ^ 2. Therefore, AC = (AB ^ 2 + BC ^ 2) ^ 0.5 = (3 ^ 2 + 4 ^ 2) ^ 0.5 = 25 ^ 0.5 = 5 cm

2) Find the median length by the formula: BD = AC / 2. Then BD = 5 cm.

2

A completely different situation arises when the median is lowered onto the legs of a right triangle. Let the triangle ABC have an angle B in a straight line, and AE and CF the medians are lowered to the corresponding legs BC and AB. Here the length of these segments is found by the formulas: AE = (2 (AB ^ 2 + AC ^ 2) -BC ^ 2) ^ 0.5 / 2

CF = (2 (BC ^ 2 + AC ^ 2) -AB ^ 2) ^ 0.5 / 2 Example: For a triangle ABC, the angle ABC is straight. The length of the leg AB = 8 cm, the angle BCA = 30 degrees. Find the lengths of the medians omitted from sharp corners.

1) Find the length of the hypotenuse AC, it can be obtained from the relation sin (BCA) = AB / AC. Hence, AC = AB / sin (BCA). AC = 8 / sin (30) = 8 / 0.5 = 16 cm.

2) Find the length of the leg of the speaker. It can be most easily found by the Pythagorean theorem: AC = (AB ^ 2 + BC ^ 2) ^ 0.5, AC = (8 ^ 2 + 16 ^ 2) ^ 0.5 = (64 + 256) ^ 0.5 = (1024) ^ 0.5 = 32 cm.

3) Find the medians from the above formulas

AE = (2 (AB ^ 2 + AC ^ 2) -BC ^ 2) ^ 0.5 / 2 = (2 (8 ^ 2 + 32 ^ 2) -16 ^ 2) ^ 0.5 / 2 = (2 (64 + 1024) -256) ^ 0.5 / 2 = 21.91 cm.

CF = (2 (BC ^ 2 + AC ^ 2) -AB ^ 2) ^ 0.5 / 2 = (2 (16 ^ 2 + 32 ^ 2) -8 ^ 2) ^ 0.5 / 2 = (2 (256 + 1024) -64) ^ 0.5 / 2 = 24.97 cm.

note

The median always divides the triangle into two other triangles, equal in area.

The intersection point of all three medians is called the center of gravity.

Useful advice

Very often, the meaning of cathetas and hypotenuses is easiest to find using trigonometric formulas.

what is the median of a rectangle