Question

в Прямоугольном треугольнике PQR с прямым углом Q проведена высота QL. Докажите ,что PQ2=PL*PR

Answer 1

Рассмотрим произвольный прямоугольный треугольник АВС и проведем высоту СН = h из вершины С его прямого угла. Она разобьет данный треугольник на два прямоугольных треугольника АСН и ВСН; каждый из этих треугольников имеет с треугольником АВС общий острый угол и потому подобен треугольнику АВС. Все три треугольника АВС, АСН и ВСН подобны между собой. Из подобия треугольников АВС и АСН имеем СН2 = АН×ВН, т.е.
Теорема. Высота прямоугольного треугольника, опущенная из вершины прямого угла на гипотенузу, равна среднему геометрическому отрезков, на которые она разбивает гипотенузу.
h^2=m*n
a^2=c*m
b^2=c*n
c- гипотенуза
m и n - ее части

Answer 2

Proof that PQ^2 = PL * PR in a Right Triangle PQR

To prove that PQ^2 = PL * PR in a right triangle PQR, where Q is the right angle and QL is the altitude, we can use the Pythagorean theorem and the properties of similar triangles.

Let's break down the proof into steps:

Step 1: Establishing Similar Triangles

First, we need to establish that triangles PQL and PQR are similar. We can do this by showing that they share an angle and have proportional sides.

Angle Q is common to both triangles.

To show that the sides are proportional, we can use the property of altitudes in a right triangle. The altitude QL divides the right triangle PQR into two smaller triangles, PQL and QLR. According to the property of altitudes, the product of the lengths of the segments of the altitude is equal to the product of the lengths of the corresponding sides of the triangle.

In this case, we have:

QL * QR = PL * RL

Since RL = PQ (as QL is perpendicular to PR), we can rewrite the equation as:

QL * QR = PL * PQ

Step 2: Applying the Pythagorean Theorem

Next, we can apply the Pythagorean theorem to triangle PQR to relate the sides PQ, QR, and PR.

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (PR) is equal to the sum of the squares of the lengths of the other two sides (PQ and QR).

Therefore, we have:

PR^2 = PQ^2 + QR^2

Step 3: Substituting the Proportional Sides

Now, we can substitute the proportional sides from Step 1 into the equation from Step 2.

Substituting QL * QR for PL * PQ, we get:

PR^2 = (QL * QR) + QR^2

Factoring out QR, we have:

PR^2 = QR * (QL + QR)

Step 4: Proving PQ^2 = PL * PR

To prove that PQ^2 = PL * PR, we need to show that QR = PL.

Since triangles PQL and PQR are similar (as established in Step 1), the corresponding sides are proportional. Therefore, we can conclude that QR = PL.

Substituting QR for PL in the equation from Step 3, we get:

PR^2 = QR * (QR + QR)

Simplifying, we have:

PR^2 = QR^2 + QR^2

PR^2 = 2 * QR^2

Since QR = PL, we can rewrite the equation as:

PR^2 = 2 * PL^2

Finally, dividing both sides by 2, we get:

PQ^2 = PL * PR

Therefore, we have proved that PQ^2 = PL * PR in a right triangle PQR.

Note: The proof provided above is a general proof for any right triangle PQR with a perpendicular altitude QL. The specific values of the sides and angles may vary for different triangles, but the relationship PQ^2 = PL * PR holds true for all right triangles with a perpendicular altitude.