Given: right triangle ABC with right angle at C, AC=22 and BC=6.Draw altitude CD where D is o hypotenuse AB. What is the ratio of the area of triangle ADC to the area of triangle CDB?
To find the ratio of the area of triangle ADC to the area of triangle CDB, we first need to determine the lengths of AD and BD.
Let's start by using the Pythagorean Theorem to find the length of AB (the hypotenuse):
AB² = AC² + BC²
AB² = 22² + 6²
AB² = 484 + 36
AB² = 520
AB = √520
AB = 2√130
Since ADC is a right triangle, and CD is the altitude, we can use the properties of similar triangles to determine the lengths of AD and BD.
The triangles ADC and ABC are similar by the AA similarity theorem since they both contain a right angle and share angle A.
Using the ratio of corresponding sides, we have:
AD / AC = CD / BC
Substituting the given values, we have:
AD / 22 = CD / 6
Simplifying, we have:
AD = (CD * 22) / 6
To find CD, we need to use the area of the triangles.
The area of triangle ADC is given by:
Area_ADC = (1/2) * AD * CD
The area of triangle CDB is given by:
Area_CDB = (1/2) * BD * CD
To find BD, we need to use the Pythagorean Theorem again:
BD² = AB² - AD²
BD² = (2√130)² - AD²
BD² = 4 * 130 - AD²
BD = √(520 - AD²)
Now we can calculate the ratio of the areas by substituting the values we found:
Ratio = Area_ADC / Area_CDB
Ratio = ((1/2) * AD * CD) / ((1/2) * BD * CD)
Ratio = AD / BD
Ratio = (CD * 22) / (√(520 - AD²))
Solving for the value of AD, we can substitute it into the ratio equation:
AD = (CD * 22) / 6
AD² = (CD * 22)² / 6²
AD² = (CD * 22)² / 36
Substituting AD² into the ratio equation:
Ratio = (CD * 22) / (√(520 - (CD * 22)² / 36))
To simplify further and calculate the exact ratio, we need to know the value of CD.