Question

Диагонали прямо угольника АВСД пересекают точке О,СД=9,АС=12 Найти периметр треугольника СОД

Answer 1

CD=9,OC=6,OD=6, P=9+6+6=21(см)

Answer 2

Given Information

We are given that the diagonals of quadrilateral ABCD intersect at point O, and that CD = 9 and AC = 12. We need to find the perimeter of triangle SOD.

Solution

To find the perimeter of triangle SOD, we need to know the lengths of its sides. Let's label the vertices of the triangle as follows: - S: the intersection of AC and BD - O: the intersection of AB and CD - D: the intersection of AD and BC

To find the length of side SO, we can use the fact that triangles SOD and SAC are similar. This is because they share an angle at O and have parallel sides (AD and BC are parallel, and AC and BD are parallel). Therefore, we can set up the following proportion:

SO/SA = SD/SC

Since we know that CD = 9 and AC = 12, we can substitute these values into the proportion:

SO/12 = SD/9

To find the length of side SD, we can use the fact that triangles SOD and BCD are similar. This is because they share an angle at D and have parallel sides (AD and BC are parallel, and AC and BD are parallel). Therefore, we can set up the following proportion:

SD/CD = BD/BC

Since we know that CD = 9, we can substitute this value into the proportion:

SD/9 = BD/BC

To find the length of side BD, we can use the fact that triangles BCD and BAC are similar. This is because they share an angle at B and have parallel sides (AD and BC are parallel, and AC and BD are parallel). Therefore, we can set up the following proportion:

BD/AC = BC/AB

Since we know that AC = 12, we can substitute this value into the proportion:

BD/12 = BC/AB

Now we have three equations with three unknowns: SO, SD, and BD. We can solve these equations simultaneously to find the lengths of the sides of triangle SOD.

Let's solve the equations step by step:

1. From the equation SO/12 = SD/9, we can rewrite it as SO = (12/9) * SD.

2. From the equation SD/9 = BD/BC, we can rewrite it as SD = (9/BC) * BD.

3. From the equation BD/12 = BC/AB, we can rewrite it as BD = (12/AC) * BC.

Now we can substitute these expressions into each other to eliminate the variables:

SO = (12/9) * ((9/BC) * BD) = (4/BC) * BD

SD = (9/BC) * BD

BD = (12/AC) * BC

Now we have expressions for SO and SD in terms of BD. We can substitute these expressions into the equation SO/12 = SD/9 to solve for BD:

((4/BC) * BD)/12 = ((9/BC) * BD)/9

Simplifying this equation, we get:

(4/BC) * BD = BD

Cancelling out BD, we get:

4/BC = 1

Solving for BC, we find:

BC = 4

Now that we know BC, we can substitute this value into the expression for BD:

BD = (12/AC) * BC = (12/12) * 4 = 4

Finally, we can substitute the values of BC and BD into the expression for SD:

SD = (9/BC) * BD = (9/4) * 4 = 9

Therefore, we have found that BD = 4, SD = 9, and BC = 4.

To find the perimeter of triangle SOD, we simply add up the lengths of its sides:

Perimeter of triangle SOD = SO + SD + OD

Since we know that SO = (4/BC) * BD = (4/4) * 4 = 4, and OD = CD = 9, we can substitute these values into the equation:

Perimeter of triangle SOD = 4 + 9 + 9 = 22

Therefore, the perimeter of triangle SOD is 22.

Answer

The perimeter of triangle SOD is 22.