in the accompanying diagram, RST is a right triangle, SU is the altitude to hypotenuse RT, RT=16 and RU=7

To solve the problem, we will use the Pythagorean theorem and the concept of similar triangles.

Step 1: Determine the length of ST:
Since RST is a right triangle, we can use the Pythagorean theorem to find the length of ST. According to the theorem, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.
Using the given information, RT = 16 and RU = 7, we can calculate the length of ST as follows:
ST^2 = RT^2 - RU^2
ST^2 = 16^2 - 7^2
ST^2 = 256 - 49
ST^2 = 207
Therefore, ST = √207.

Step 2: Determine the length of SU:
We know that SU is the altitude to hypotenuse RT. Since SU and UT are altitudes to two different triangles, the triangles RST and RSU are similar. Therefore, we can use the concept of similar triangles to find the length of SU.
By the similarity of triangles RST and RSU, we can set up the proportion of their corresponding sides:
ST/SR = SU/RT
ST/16 = SU/16
Since ST = √207, we can substitute this value in the equation:
√207 / 16 = SU / 16
Simplifying, we get:
√207 = SU
Therefore, SU = √207.

So, the length of ST is √207 and the length of SU is also √207.

To find the length of the altitude SU, we can use the concept of similar triangles.

In the given right triangle RST, we have RT = 16 and RU = 7.

We know that in a right triangle, the altitude drawn from the right angle to the hypotenuse creates two smaller similar triangles.

Let's denote the length of SU as 'x'.

Now, we can set up a proportion between the two similar triangles:

RS/SU = RT/RU

Using the given values, we have:

16/(16 - x) = 7/x

Now, cross-multiply to solve for 'x':

16 * x = 7 * (16 - x)

16x = 112 - 7x

Adding 7x to both sides:

16x + 7x = 112

Combine like terms:

23x = 112

Divide both sides by 23:

x = 112/23

So, the length of altitude SU is approximately 4.87 units.

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