In the diagram below of right triangle ACB, altitude CD intersects AB at D. If AD=3 and DB=4, find the length of CD in simplest radical form.
How do I go about figuring out this problem?
Please help. thanks
To solve this problem, you can use the Pythagorean theorem and the concept of similar triangles.
Here are the steps you can follow:
Step 1: Label the given lengths and the length you are trying to find in the diagram. Let's label the length of CD as x.
Step 2: Since AD and DB are given, you can find the length of AB using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (AB in this case) is equal to the sum of the squares of the lengths of the other two sides (AD and DB).
So, AB^2 = AD^2 + DB^2
Substituting the given values, you get:
AB^2 = 3^2 + 4^2
AB^2 = 9 + 16
AB^2 = 25
Taking the square root of both sides, you find:
AB = √25
AB = 5
Step 3: Now that you have the lengths of AC (x) and AB (5), you can find the ratio between the corresponding sides of the two similar triangles ACD and ACB. Both triangles share the same angle at C. The ratio of the lengths of corresponding sides of similar triangles is equal.
Using the ratio of the corresponding sides, you can set up the following proportion:
AC/AB = CD/AD
Substituting the given and known values, you get:
x/5 = CD/3
Step 4: Solve the proportion for CD. Cross-multiplying, you have:
3x = 5 * CD
Dividing both sides by 3, you find:
CD = (5/3) * x
Step 5: Finally, substitute the known value of AB (5) in place of CD:
CD = (5/3) * x = 5
So, the length of CD is 5.
To summarize:
1. Use the Pythagorean theorem to find the length of AB.
2. Set up a proportion to relate the lengths of AC, AB, CD, and AD.
3. Solve the proportion for CD by cross-multiplying.
4. Substitute the known value of AB to solve for CD.
* The altitude to the hypotenuse of a right triangle creates two similar triangles, each similar to the original right triangle and to each other.
* The altitude to the hypotenuse of a right triangle is the geometric mean between the segments of the hypotenuse created by the point where the altitude intersects the hypotenuse or h^2 = xy.
* With an altitude drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse in contact with the leg.