In the figure, is a trapezoid with bases and . Its diagonals intersect at . The numbers inside two of the triangular regions represent their areas, in square units.
Which choice is the area, in square units, of trapezoid ?
When diagonals divide a trapezoid into four triangles, first look for similar triangles formed near the parallel bases. Use their area ratio to find a ratio along a diagonal, then compare triangles that share a height. This converts the diagram into simple proportional area relationships before adding the regions.
Hints
Compare the upper and lower triangles
Use the parallel bases and the intersecting diagonals to identify a pair of similar triangles.
Connect area ratio to side ratio
For similar triangles, the ratio of their areas is the square of the ratio of corresponding side lengths.
Use shared heights
Each unmarked triangle shares a height with one labeled triangle. Compare their bases along diagonal .
Desmos Guide
Use the area ratio
An algebraic approach is faster here. For a numerical check in Desmos, enter sqrt(18/50) to find the ratio of corresponding side lengths for the similar triangles.
Calculate the unmarked areas
Enter 18/(sqrt(18/50)) and 50*(sqrt(18/50)). These calculations verify that the two unmarked triangular regions have equal areas.
Add all regions
Enter 18+50+2(30) to verify the total area of the trapezoid.
Step-by-step Explanation
Use the similar triangles
Because is parallel to , triangles and are similar. Their area ratio is
Therefore, the ratio of corresponding side lengths is , so .
Find the area of a side region
Triangles and have bases and on the same line and share the same height from point . Therefore, their areas have ratio .
Since triangle has area , triangle has area
Find the other side region and total area
Triangles and share the same height from point , and their bases are and . Their area ratio is also , so triangle has area
Add the areas of all four triangular regions: