Question

В треугольнике KLM со сторонами KL = 2021, KM = 2022 проведена биссектриса угла K. На эту биссектрису опущен перпендикуляр LH. Найдите FH, где F – середина LM.

Answer 1

Ответ на фото............

Answer 2

Problem Analysis

We are given a triangle KLM with sides KL = 2021 and KM = 2022. A bisector of angle K is drawn, and a perpendicular LH is dropped onto this bisector. We need to find FH, where F is the midpoint of LM.

Solution

To find FH, we can use the properties of similar triangles. Let's denote the length of FH as x.

Since F is the midpoint of LM, we can conclude that LF = FM = x.

Now, let's consider triangles KLF and KHM. These triangles are similar because they share angle K and have a pair of proportional sides (LF and HM).

Using the property of similar triangles, we can set up the following proportion:

KL / KH = LF / HM

Substituting the given values, we have:

2021 / KH = x / 2022

To find FH, we need to find x. Let's solve the equation for x:

x = (2021 * 2022) / KH

To find KH, we can use the angle bisector theorem. According to this theorem, the ratio of the lengths of the segments formed by an angle bisector in a triangle is equal to the ratio of the lengths of the opposite sides.

In triangle KLM, the angle bisector LH divides the side KL into two segments, KH and HL. Since KL = 2021, we can write:

KH / HL = KM / LM

Substituting the given values, we have:

KH / HL = 2022 / (2x)

Simplifying the equation, we get:

KH = (2022 * HL) / (2x)

Now, let's substitute this value of KH into the equation for x:

x = (2021 * 2022) / [(2022 * HL) / (2x)]

Simplifying further, we have:

x^2 = (2021 * 2022 * 2x) / (2022 * HL)

Simplifying again, we get:

x^2 = 4042x / HL

Multiplying both sides by HL, we have:

x^2 * HL = 4042x

Dividing both sides by x, we get:

x * HL = 4042

Finally, solving for x, we have:

x = 4042 / HL

Therefore, FH is equal to 4042 divided by the length of HL.

Answer

The length of FH, where F is the midpoint of LM, can be found by dividing 4042 by the length of HL.