Question

Выпуклом четырехугольнике ABCD диагональ AC является биссектрисой угла DAB и пересекает диагональ BD в точке K. Найдите BC, если известно что АК= 9 ,КС=3 и около четырехугольника ABCD можно описать окружность.Помогите решить плиз, желательно с объяснением

Answer 1

Площади треугольников ABC и CDB равны между собой.
A и D находятся на равном расстоянии от BC и AD параллельна BC.
Треугольник ABC - равносторонний, т. к. уголы BCA, CAD и BAC равны.
Так что BC = AB = 4.

Answer 2

Given Information:

We are given that in convex quadrilateral ABCD, diagonal AC is the bisector of angle DAB and intersects diagonal BD at point K. We also know that AK = 9, KC = 3, and a circle can be circumscribed around quadrilateral ABCD.

Solution:

To find BC, we can use the properties of a cyclic quadrilateral and the intersecting diagonals.

Properties of a Cyclic Quadrilateral:

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. It has the following properties: 1. Opposite angles are supplementary. 2. The sum of the measures of any pair of opposite angles is 180 degrees.

Using the Properties of a Cyclic Quadrilateral:

Since a circle can be circumscribed around quadrilateral ABCD, we can use the properties of a cyclic quadrilateral to find the value of BC.

1. From property 1, we know that angle ABC + angle CDA = 180 degrees. 2. From property 2, we know that angle ABC + angle BCD = 180 degrees.

Combining these two equations, we can write: angle ABC + angle CDA = angle ABC + angle BCD

Since angle CDA and angle BCD are the same angle, we can simplify the equation to: angle ABC = angle BCD

Using the Bisector Property:

We are given that diagonal AC is the bisector of angle DAB. This means that angle DAC is equal to angle CAB.

Applying the Angle Bisector Theorem:

The Angle Bisector Theorem states that in a triangle, a line segment bisecting an angle divides the opposite side into segments that are proportional to the lengths of the other two sides.

In triangle AKC, we have: AK/CK = AB/BC

Substituting the given values, we get: 9/3 = AB/BC 3 = AB/BC

Using the Proportional Relationship:

From the previous equation, we have: 3 = AB/BC

Multiplying both sides by BC, we get: 3BC = AB

Applying the Angle Bisector Property:

We know that angle ABC = angle BCD. Since angle ABC = angle BCD, we can substitute angle ABC with angle BCD in the equation 3BC = AB.

3BC = BC

Dividing both sides by BC, we get: 3 = 1

This equation is not possible, as 3 cannot be equal to 1. Therefore, there is no solution for BC that satisfies the given conditions.

Please note that there might be a mistake in the given information or calculations. Double-check the given information and calculations to ensure accuracy.