The altitude to the hypotenuse of a right triangle ABC divides the hypotenuse into 12mm and 16mm segments. Find the lengths of each of the following.

A. The altitude to the hypotenuse
B. The shorter leg of angle ABC
C. The longer leg of angle ABC

To solve this problem, we can use the property of similar triangles. Let's label the three sides of the right triangle ABC as follows:

AC: The hypotenuse of the triangle
BC: The shorter leg (adjacent to angle ABC)
AB: The longer leg (opposite angle ABC)

We know that the altitude to the hypotenuse divides the hypotenuse into two segments, with lengths of 12mm and 16mm.

Now let's solve each part of the question:

A. To find the altitude to the hypotenuse, we need to determine the length of the segment perpendicular to the hypotenuse. Let's call this length "h" mm.

Using similar triangles, we can set up the following proportion:

h / 12 = 16 / AC

Cross-multiplying and rearranging the equation, we get:

h = (16 * 12) / AC

B. To find the shorter leg of angle ABC (BC), we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

So, we have:

BC^2 + h^2 = 12^2

Substituting the value of h from equation (A):

BC^2 + [(16 * 12) / AC]^2 = 12^2

C. To find the longer leg of angle ABC (AB), we can use a similar process. Again, using the Pythagorean theorem:

AB^2 + h^2 = 16^2

Substituting the value of h from equation (A):

AB^2 + [(16 * 12) / AC]^2 = 16^2

By solving the above equations, you will be able to find the values of h, BC, and AB, which represent the lengths of the altitude, the shorter leg, and the longer leg, respectively, in the right triangle ABC.

To find the lengths of the desired segments, we can use the concept of similar triangles. Let's start with the given information:

Given:
- The altitude to the hypotenuse divides it into segments of 12mm and 16mm.

Let's label the points:
- The altitude from vertex B to the hypotenuse is point D.
- The two segments created on the hypotenuse are BD = 12mm and DC = 16mm.

Now, we'll find the lengths of each of the desired segments step-by-step:

A. The altitude to the hypotenuse:
Since the altitude to the hypotenuse divides the right triangle into two smaller similar triangles, we can use their similarity to find the length of the altitude.

Applying the concept of similarity:
In triangle ABC and triangle BDC, we have the following ratios:
1. BD/AB = DC/BC (The segments on the hypotenuse divide it proportionally).
2. BD/AD = DC/CD (The altitude divides the triangles proportionally).

From equation 1, we can rewrite it as BD/AB = DC/BC as BD/12 = 16/BC.

Now, let's solve for BC:
Cross-multiplying, we get BD * BC = AB * DC.
Substituting the given values, 12 * BC = AB * 16.

From equation 2, BD/AD = DC/CD.
Substituting our known values, 12/AD = 16/CD.

Now, let's solve for AD:
Cross-multiplying, we get 12 * CD = AD * 16.

Based on these two equations, we can equate AB * DC and AD * BC since they're both equal to 192 (12 * 16).

BC = AB * DC / 12 = AB * 16 / 12 = 4/3 * AB,
AD = CD * 12 / 16 = CD * 3 / 4 = 3/4 * CD.

So, BC = 4/3 * AB and AD = 3/4 * CD.

B. The shorter leg of angle ABC:
From the similarity of triangles ABC and BDC, we can set up the following proportion: AB/BC = BC/BD.
Substituting the known values: AB/BC = BC/12.
Cross-multiplying, we get AB * 12 = BC^2, or 4/3 * AB * 12 = BC^2, which simplifies to 16AB = BC^2.

Since AB = BC * 3/4 (from the length of AD), we can substitute it in the equation above to get:
16AB = (BC * 3/4)^2 = 9/16 * BC^2.
Therefore, 16AB = 9/16 * BC^2. Cross-multiplying, we have 16AB * 16/9 = BC^2.

So, AB = BC^2 * 16/9.

C. The longer leg of angle ABC:
From the similarity of triangles ABC and BDC, we have AC/DC = BC/BD.
Substituting the known values: AC/16 = BC/12.
Cross-multiplying, we get AC * 12 = BC * 16, or AC = BC * 16/12.

Now, let's summarize the results:
A. The altitude to the hypotenuse: AD = 3/4 * CD.
B. The shorter leg of angle ABC: AB = BC^2 * 16/9.
C. The longer leg of angle ABC: AC = BC * 16/12, or AC = 4/3 * BC.

Now, solving for BC will allow us to find the values for AD, AB, and AC.