in right triangle ABC, the altitude CD is drawn to hypotynuse AB. The length of DB is 5 units longer than AD.
a) If AD = x, express DB in terms of x.
b) If CD = 6, write and equation in terms of x to find AD.
c) Find AD.
d) Find AC.
To find the answers, we will use the properties of right triangles and proportions. Let's go step by step:
a) If AD = x, we know that DB is 5 units longer than AD. Therefore, we can express DB in terms of x as DB = x + 5 units.
b) We are given that the length of CD (the altitude) is 6 units. Since CD is perpendicular to AB, we know that triangles ACD and BCD are similar to triangle ABC. Hence, we can set up proportions to determine the relationship between the sides.
Considering the sides of triangle ABC, we have:
AC/AB = CD/AC
Substituting the given values, we have:
AC/AB = 6/AC
c) To find AD, we can use the Pythagorean Theorem in triangle ACD. According to the theorem, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In this case, we have:
AD^2 + CD^2 = AC^2
Substituting the known values, we get:
x^2 + 6^2 = AC^2
d) To find AC, we can substitute the value of DB from part a) into the equation from part c):
(DB - 5)^2 + CD^2 = AC^2
Substituting the known values, we get:
(x + 5 - 5)^2 + 6^2 = AC^2
Simplifying:
x^2 + 36 = AC^2
So, the equation to find AC is x^2 + 36 = AC^2.
Now, to find the specific values of AD and AC, we need more information or values for x or the triangle's dimensions.