Question

Сумма гипотенузы АС и катета АВ прямоугольного треугольника АВС равна 21см, а их разность равна 5 см.Найдите расстояние от вершины А до прямой ВС. Постройте равнобедренный треугольник по основанию и медиане, проведенной к основанию!

Answer 1

В прямоугольном Δ АВС гипотенуза АС = х, катет АВ= у
х + у = 21
х - у = 5
Решим эту систему. Сложим эти 2 уравнения, получим:
2х = 26
х = 13 (АВ)
х + у = 21
13 + у = 21
у = 21 - 13 = 8 (АС)
расстояние от А до прямой ВС - это расстояние АВ. АВ =  8

Answer 2

Problem Statement

We are given a right triangle ABC, where the sum of the hypotenuse AC and the leg AB is 21 cm, and their difference is 5 cm. We need to find the distance from vertex A to the line BC. Additionally, we are asked to construct an isosceles triangle with the given base and the median drawn to the base.

Solution

To solve this problem, let's first assign variables to the given lengths: - Let AC be x cm. - Let AB be y cm.

According to the problem statement, we have the following equations: 1. x + y = 21 cm2. x - y = 5 cm We can solve this system of equations to find the values of x and y.

Adding equations 1 and 2, we get: x + y + x - y = 21 + 5 2x = 26 x = 13 cm

Substituting the value of x into equation 1, we can find y: 13 + y = 21 y = 8 cm

Therefore, AC = 13 cm and AB = 8 cm.

Distance from Vertex A to Line BC

To find the distance from vertex A to line BC, we need to draw a perpendicular line from A to BC. Let's call the point where the perpendicular intersects BC as D.

Since triangle ABC is a right triangle, we can use the Pythagorean theorem to find the length of BC. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Applying the Pythagorean theorem to triangle ABC, we have: AC^2 = AB^2 + BC^2

Substituting the values of AC and AB, we get: 13^2 = 8^2 + BC^2 169 = 64 + BC^2 BC^2 = 169 - 64 BC^2 = 105 BC = √105 cm

Now, let's find the distance from vertex A to line BC. This distance is equal to the length of the perpendicular AD.

To find AD, we can use the formula for the area of a triangle. The area of a triangle can be calculated as half the product of the base and the height.

The area of triangle ABC can be calculated as: Area = (1/2) * AB * BC

Substituting the values of AB and BC, we get: Area = (1/2) * 8 * √105 Area = 4 * √105

Now, let's find the height of triangle ABC, which is equal to AD. We can rearrange the formula for the area of a triangle to solve for the height: Height = (2 * Area) / Base

Substituting the values of Area and Base, we get: Height = (2 * 4 * √105) / 8 Height = √105 / 2

Therefore, the distance from vertex A to line BC is √105 / 2 cm.

Construction of an Isosceles Triangle

To construct an isosceles triangle with the given base AB and the median drawn to the base, we can follow these steps: 1. Draw a line segment AB of length 8 cm. 2. Bisect AB to find the midpoint M. 3. Draw a line segment MC perpendicular to AB, passing through M. 4. Extend MC to a point D such that MD = √105 / 2 cm. 5. Connect AD and BD to form triangle ABD.

Triangle ABD is an isosceles triangle with base AB and the median AD drawn to the base.

Note: The construction assumes that the given base AB is the base of the isosceles triangle and the median AD is drawn to the base.

I hope this helps! Let me know if you have any further questions.