The bases of trapezoid ABCD are \overline{AB} and \overline{CD}. Let P be the intersection of diagonals \overline{AC} and \overline{BD}. If the areas of triangles ABP and CDP are 8 and 18, respectively, then find the area of trapezoid ABCD.
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To find the area of trapezoid ABCD, we can add the areas of triangles ABP and CDP.
Let's denote the length of base AB as b and the length of base CD as a.
The formula to find the area of a triangle is (base * height) / 2.
The given areas of triangles ABP and CDP are 8 and 18, respectively. So, we have:
Area of triangle ABP = (b * heightABP) / 2 = 8 Eq. (1)
Area of triangle CDP = (a * heightCDP) / 2 = 18 Eq. (2)
Now, let's find the ratio of the areas of triangles ABP and CDP:
8 / 18 = (b * heightABP) / (a * heightCDP)
Simplifying this equation, we get:
4 / 9 = (b * heightABP) / (a * heightCDP) Eq. (3)
We know that the diagonals of a trapezoid divide each other proportionally. By using similar triangles, we can say that:
heightABP / heightCDP = AP / CP
Using the properties of the intersection point P, we have:
AP / CP = BP / DP Eq. (4)
Since AP + BP = AB and CP + DP = CD (as the diagonals divide each other proportionally), we can rewrite Eq. (4) as:
AP / CP = (AB - AP) / (CD - CP)
Cross-multiplying and simplifying, we get:
AP * (CD - CP) = CP * (AB - AP)
AP * CD - AP * CP = CP * AB - CP * AP
AP * CD = CP * AB
AP / AB = CP / CD Eq. (5)
Substituting Eq. (5) into Eq. (4), we have:
AP / CP = CP / CD
Taking the square root of both sides, we get:
sqrt(AP / CP) = sqrt(CP / CD)
Simplifying, we have:
sqrt(AP / CP) = CP / CD
Cross-multiplying, we get:
(AP / CP) * (CP / CD) = 1
Simplifying further, we have:
AP / CD = 1 Eq. (6)
Now, let's substitute Eq. (6) into Eq. (3):
4 / 9 = (b * heightABP) / (a * heightCDP)
8 / 9 = (b * heightCD) / (a * heightCDP)
Cross-multiplying, we have:
8 * (a * heightCDP) = 9 * (b * heightCD)
Simplifying and rearranging the equation, we get:
a * heightCDP = (9/8) * b * heightCD
Dividing both sides by b * heightCD, we get:
(a * heightCDP) / (b * heightCD) = 9/8
Now, let's denote the height of the trapezoid as h. Since heightCDP and heightCD are both heights of the trapezoid, we can rewrite the equation as:
(a * h) / (b * h) = 9/8
Simplifying, we have:
a / b = 9 / 8
Now, we know that the area of a trapezoid is given by the formula:
Area = [(a + b) * h] / 2
Substituting the values of a/b = 9/8, we have:
Area = [(9b/8 + b) * h] / 2
Simplifying further, we get:
Area = [((9+8)/8) * b * h] / 2
Area = (17/8) * b * h
Therefore, the area of trapezoid ABCD is (17/8) * b * h.