A square is constructed using the hypotenuse line AC of right triangle ABC as a side, as shown below. Find the area of the square if AB = 5 and BC = 9.
I tried to approach this by finding the hypotenuse, which i found was 10, then i used that to find the area of the square which was 100, this was wrong. Then i went and did the same process with 11 and 121, but it was also wrong. What did i do wrong, and what should i be doing instead?
First of all , the length of the hypotenuse is not 10
AC^2 = 5^2 + 9^2 = 106
So AC = √106
Area of square on that side = AC*AC = AC^2 = 106
- end of problem!
Since I don't see your diagram, I have no idea where your 11 and 121 comes from.
To find the area of the square constructed on the hypotenuse line AC of right triangle ABC, you need to use the Pythagorean theorem.
The Pythagorean 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 lengths of the other two sides.
In this case, you have AB = 5 and BC = 9. To find the length of AC, you can use the Pythagorean theorem:
AC^2 = AB^2 + BC^2
AC^2 = 5^2 + 9^2
AC^2 = 25 + 81
AC^2 = 106
AC ≈ √106 ≈ 10.29
Since the square is constructed using the length of AC as a side, the area of the square would be (AC)^2:
Area of the square = (10.29)^2 = 105.96
So, the correct area of the square is approximately 105.96 square units.
Therefore, you made an error by considering the hypotenuse as 10 or 11. The correct length of the hypotenuse is approximately 10.29, which should be squared to find the area of the square.
To find the area of the square, you need to be aware of a key property of squares: all sides of a square are equal in length. In this case, the hypotenuse line AC is used as one side of the square.
Let's go through the steps again and see where the mistake might have occurred:
1. Start with the given information. AB = 5 and BC = 9. These are the lengths of two sides of the right triangle ABC.
2. Use the Pythagorean theorem to find the length of the hypotenuse AC. The Pythagorean 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 lengths of the other two sides. In this case, it would be AC^2 = AB^2 + BC^2.
AC^2 = 5^2 + 9^2
AC^2 = 25 + 81
AC^2 = 106
AC ≈ √106
Here, we find that the length of the hypotenuse AC is approximately equal to the square root of 106.
3. Now, to find the area of the square, you need to square the length of the side (AC). Since all sides of a square are equal, the area is equal to the square of any of its sides.
Area of the square = AC^2
Substituting the value of AC obtained from step 2:
Area of the square ≈ (√106)^2
Area of the square ≈ 106
Therefore, the area of the square is approximately equal to 106 square units.
It seems that you were making the mistake of using the hypotenuse directly as the side length of the square, instead of finding the length of the hypotenuse using the Pythagorean theorem and then squaring it to get the area.
Note: It is worth mentioning that the exact value of the side length of the square and its area might involve irrational numbers because we used the square root of 106. If you were asked to find an exact value in simplified form, you would need to leave it as (√106)^2 or 106, depending on the level of detail required.