In right triangle ABC, we are given that ∠ABC=90∘ and AC=34. D is a point on line segment BC such that BD=12,DC=18. What is the length of AD?
from the information,
BC = 30
so AB^2 + 30^2 = 34^2
AB = 16
Then in triangle ABC
AD^2 = 16^2 + 12^2 = 400
AD = √400 = 20
math sucks
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In triangle ABC, AC is the hypotenuse, and we are given that its length is 34. Let's label the length of side AB as a and the length of side BC as b.
According to the Pythagorean theorem, we have the equation:
AC^2 = AB^2 + BC^2
Replacing the variables with the given values, we get:
34^2 = a^2 + b^2
Simplifying, we have:
1156 = a^2 + b^2
Now, let's consider triangle ABD. We know that BD = 12. We want to find the length of AD, which we can label as x.
Using the Pythagorean theorem again, we have:
AD^2 = AB^2 + BD^2
Substituting the given values, we have:
x^2 = a^2 + 12^2
Simplifying, we have:
x^2 = a^2 + 144
Since we have two equations, we can solve them simultaneously to find the values of a and x.
Subtracting the second equation from the first equation, we get:
a^2 + b^2 - (a^2 + 144) = 1156 - 144
Simplifying, we have:
b^2 = 1012
Taking the square root of both sides, we find:
b ≈ 31.79
Now, substituting the value of b back into the first equation, we have:
a^2 + 31.79^2 = 1156
Simplifying, we have:
a^2 ≈ 344.72
Taking the square root of both sides, we find:
a ≈ 18.57
Therefore, the length of AD is approximately 18.57.