find the area of the trapezoid whose diagonals are 30 each and height of 18?
I spent 2 hrs and I can't figure it out Pls. help
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Who says this trapezoid is not a rectangle with height 18 and base sqrt (30^2-18^2)?
1. extend the top line of the trapezoid to the left, to make a right triangle with the top line and the diagonal.
2. This triangle has side of 18 and hypotenuse of 30. By pythagorean theorem, the extended top line is 24.
3. Draw a line from upper right point of trapezoid to base, making a another 18x24x30 triangle.
4. The area of the trapezoid is 18x24 = 432, which is the area of the rectangle formed from the 2 triangles.
5. The small triangles (one inside the rectangle upper left and one outside of the rectangle lower left) are congruent.
To find the area of a trapezoid, you can use the formula:
Area = ((b1 + b2) * h) / 2
Where:
- b1 and b2 are the lengths of the bases of the trapezoid
- h is the height of the trapezoid
In this case, you mentioned that the diagonals of the trapezoid are each 30 units long and the height is 18 units.
To find the lengths of the bases (b1 and b2), you can use the Pythagorean theorem. The diagonals of a trapezoid are typically not enough information to directly determine the lengths of the bases. However, since you mentioned that the diagonals are both 30 units long and that the height of the trapezoid is 18 units, we can determine the lengths of the bases.
Let's label the trapezoid as ABCD, with diagonal AC = 30 units, diagonal BD = 30 units, and height h = 18 units.
Since the diagonals of a trapezoid bisect each other, this means that the midpoint (M) of AC and BD is the same point. Let's use this information to split the trapezoid into two right triangles, AMC and BMD.
Using the Pythagorean theorem, we can find the length of AM (which is also the length of MC) and the length of BM (which is also the length of MD).
Let's calculate:
AM = MC = √(AC² - h²)
AM = MC = √(30² - 18²)
AM = MC = √(900 - 324)
AM = MC = √576
AM = MC = 24 units
BM = MD = √(BD² - h²)
BM = MD = √(30² - 18²)
BM = MD = √(900 - 324)
BM = MD = √576
BM = MD = 24 units
Now that we know the lengths of AM (MC) and BM (MD), we can find the lengths of the bases (b1 and b2) by adding twice the lengths of AM and BM to the lengths of the diagonals:
b1 = AC + 2 * AM
b1 = 30 + 2 * 24
b1 = 30 + 48
b1 = 78 units
b2 = BD + 2 * BM
b2 = 30 + 2 * 24
b2 = 30 + 48
b2 = 78 units
Now we have the lengths of the bases (b1 and b2) and the height (h), so we can use the area formula:
Area = ((b1 + b2) * h) / 2
Area = ((78 + 78) * 18) / 2
Area = (156 * 18) / 2
Area = 2808 / 2
Area = 1404 square units
Therefore, the area of the trapezoid is 1404 square units.