Lines L1 and L2 intersect at O. A and C are points on L1 such that OA=11,OC=9. B and D are points on L2 such that OB=3,OD=22. What is the ratio [OCD]/[OAB]?
To find the ratio [OCD]/[OAB], we first need to find the areas of triangles OCD and OAB.
Step 1: Find the length of AD.
We can find the length AD by using the triangle proportionality theorem. Since OB is to OD as OA is to AD, we have:
OB/OD = OA/AD
Substituting the given values, we get:
3/22 = 11/AD
Cross-multiplying gives:
3 * AD = 11 * 22
AD = (11 * 22) / 3
AD = 242 / 3
Step 2: Find the length of BC.
Similarly, using the triangle proportionality theorem, we can find the length BC. Since OC is to OA as OB is to BC, we have:
OC/OA = OB/BC
Substituting the given values, we get:
9/11 = 3/BC
Cross-multiplying gives:
9 * BC = 11 * 3
BC = (11 * 3) / 9
BC = 33 / 9
BC = 11 / 3
Step 3: Find the areas of triangles OCD and OAB.
The area of triangle OCD can be found using the formula for the area of a triangle:
Area(OCD) = (1/2) * base * height
We know that the base OC has a length of 9 and the height AD has a length of 242/3. Substituting into the formula, we get:
Area(OCD) = (1/2) * 9 * (242/3)
Area(OCD) = 121 * (242/3)
Area(OCD) = 29746/3
Similarly, we can find the area of triangle OAB:
Area(OAB) = (1/2) * base * height
The base OA has a length of 11 and the height BC has a length of 11/3. Substituting into the formula, we get:
Area(OAB) = (1/2) * 11 * (11/3)
Area(OAB) = 121/3
Step 4: Find the ratio [OCD]/[OAB].
The ratio [OCD]/[OAB] is the ratio of the areas of triangles OCD and OAB:
[OCD]/[OAB] = Area(OCD) / Area(OAB)
Substituting the previously calculated values, we get:
[OCD]/[OAB] = (29746/3) / (121/3)
[OCD]/[OAB] = (29746/3) * (3/121)
[OCD]/[OAB] = 29746/121
Therefore, the ratio [OCD]/[OAB] is 29746/121.