Pythagorean Theorem By: Mona

Our goal is to find the length of line AB.

To start we are going to find the length of line BD. Then using the Pythagorean Theorem we can use this measurement to find the length of line AB.

Labelling the Diagram

I labelled line DC a, in yellow, and line BD b, in pink. However, you could label it the other way around if you wanted to. Since they are both the legs of the right-angled triangle it doesn't matter which leg is a or b.

I labelled line BC c, in green. Since this line is the hypotenuse, (The longest line in a right-angled triangle, which is ALWAYS across from the right angle) it must be labelled c when using the Pythagorean Theorem.

There are different ways to do this, but to figure out the line measurement of line BD I am going to use the formula for the Pythagorean Theorem.

Step 1

The first step is to replace the variables with the information we know. For example a^2 becomes (3u)^2. This is because line DC or line a (same line) Is 3 units long. B^2 stays the same because we don't know it's measurement yet.

Tip: The 3 and the u go inside the brackets because other wise it would be read as 3 sauare units, instead of 3 unit squared.

Step 2

Next, following the order of operations (BEDMAS) we have to square everything that has to be squared. For example (3u)^2 becomes 9u^2. This is because 3 * 3 = 9. b^2 still stays the same because we still don't know what it is.

TIP: The brackets are now gone because if you square a measurement of units, they become square units. The square sign does not mean you have to square the 9 again.

Tip: 6.71 ^2 = 45.0241. I rounded this number to 45. Because I rounded it, I had to put a little dot over the equal sign showing it is a rounded number.

Step 3

Now we have all the information to figure out the measurement of b^2. If we know that 9 + b = 44, that means that 44 - 9 = b. On the left side of the equal sign we want it to show just b^2. If it's adding 9u, then if we take away 9u, it's left with just b^2. We took away 9u on the left side so we have to take away 9u on the right side because of the balance model rule. The formula is basically now showing b^2 = 45u - 9u. Underneath we have to follow the order of operations. Once we do that we are left with the measurement for b^2.

Tip: Because in the previous step we rounded to 45, we have to keep that dot over the equal sign because now we are finishing the operations with the rounded number. We have to leave the dot over the equal sign for the rest of the operations.

Step 4

So now we know the measure of line BD squared. But we want to know just the line measure. So we have to squared root our answer to get just the line measure.

Tip: If you square root a square (b^2 square root) they cancel each other out, leaving the original number. Also if you square root a square unit, it goes back to the original unti.

Now we know that the measure of line BD (or b) is 6u because 6 * 6 = 36.

Everything put together

Now that we know the measure of line BD we can find the measure of line AB using the Pythagorean Theorem.

Labelling the Diagram Part 2

I left line BD labeled at b in pink again to make it less confusing. That leaves line AD as a which I coloured as yellow again. Once again, you could label it the other way around as it doesn't really matter.

Line AB is the hypotenuse so I had to label it c, and I coloured it green again.

Now that we know the measurements of the two legs we can use this information to find the measurement of the hypotenuse. We are going to use the Pythagorean Theorem.

Step 1

First we need to fill in the variables with the measurements we know. This time it's c^2 that is the unknown, so we leave it as c^2. Once we have the numbers, following the order of operations we square them.

Tip: Because in the last step we rounded a number and we are still using that number the dot over the equal sign has to stay for this set of operations as well. The dot doesn't need to be there for the formula at the beginning (a^2 + b^2 = c^2).

Tip: Like last time we need to have the u for unit inside the brackets with the measurement until we square it.

Step 2

The next step in BEDMAS is addition. Once we've added the the squared numbers you have the square of the hypotenuse.

Step 3

Now we know the square of line AB. However, we only want the measure of the line. This means we have to square root it.

Tip: The square root of 61 is 7.810249...... I rounded it to 7.8.

Everything put together

The measurement of line AB is 7.8u.

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