right triangle ABC with right angle C, AC=22 and BC=6, altitude CD where D is on hypotenuse AB. What is the ratio of the area of triangle ADC to the area of triangle CDB? Write ratio as n:1 and round n to the nearest hundredth.

Mark you similar triangles filling in all given information

I see
triangle ADC similar to
triangle CDB

The ratio of areas of two triangles is equal to the squares on their corresponding sides.

Triangle ADC : triangle CDB = (CA)^2 : (BC)^2
= 22^2 : 6^2
= 484 : 36
= 121 : 9
or (121/9) : 1

n = 121/9 or 13 4/9 or 13.44444.....

To find the ratio of the area of triangle ADC to the area of triangle CDB, we first need to find the lengths of AD and DB.

Since triangle ABC is a right triangle, we can use Pythagoras' theorem to find the length of AB. The theorem states that the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AC and BC) in a right triangle.

Using Pythagoras' theorem:
AB^2 = AC^2 + BC^2
AB^2 = 22^2 + 6^2
AB^2 = 484 + 36
AB^2 = 520
AB = √520

Now, to find the lengths AD and DB, we can use similar triangles. In this case, triangles ADC and CDB are similar because they share angle C.

Let's denote the length of AD as x, then the length of DB would be (AB - x).

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

AD/CD = CD/DB

Substituting the lengths:
x/CD = CD/(AB - x)

Cross-multiplying, we get:
CD^2 = x * (AB - x)

Now, we have two variables in the equation: CD and x. To find the value of x, we need to use the fact that the length of CD is an altitude of the triangle.

The area of triangle ABC can be calculated using the formula:
Area = (1/2) * base * height

Substituting the values:
Area of triangle ABC = (1/2) * AB * CD

Since CD is an altitude of triangle ABC, it can be written as:
Area of triangle ABC = (1/2) * AB * AD

Substituting AB value, we get:
Area of triangle ABC = (1/2) * √520 * AD

Now, we know that the area of triangle ADC is (1/2) * AD * CD, and the area of triangle CDB is (1/2) * (AB - x) * CD.

To find the ratio of the area of triangle ADC to the area of triangle CDB, we can divide the area of triangle ADC by the area of triangle CDB and round it to the nearest hundredth.

(Area of triangle ADC) / (Area of triangle CDB) = [(1/2) * AD * CD] / [(1/2) * (AB - x) * CD]
= (AD * CD) / ((AB - x) * CD)
= AD / (AB - x)

Now, let's substitute the values and calculate the ratio:

x = AD / (AB
x = AD / (√520

Therefore, the ratio of the area of triangle ADC to the area of triangle CDB, rounded to the nearest hundredth, is AD/(AB - AD) : 1, or approximately 0.41 : 1.