Home » Test Prep » ACT Section 2: Math » Which of the following is equal to the shaded area?
In the diagram above, ABCD is a square with an area of 100 cm2 and lines BD and AC are the diagonals of ABCD. If line EF is parallel to line BC and the length of line CF = 3v2 cm, which of the following is equal to the shaded area?
A. 25 cm2
B. 39 cm2
C. 64 cm2
D. 78 cm2
E. 89 cm2
Correct Answer: B
Explanation/Reference:
Explanation:
The area of a square is equal to S2 where S is the length of one side of the square. A square with an area of 100 cm2 has sides that are each equal to √100 = 10 cm. The diagonal of a square is equal to √2 times the length of a side of the square. Therefore, the lengths of diagonals AC and BD are 10v2 cm. Diagonals of a square bisect each other at right angles, so the lengths of segments OB and OC are each 5√2 cm. Since lines BC and EF are parallel and lines OC and OB are congruent, lines BE and CF are also congruent. The length of line OF is equal to the length of line OC plus the length of line CF: 5√2 + 3v2 = 8v2 cm. In the same way, OE = OB + BE = 5√2 + 3√2 = 8 √2 cm. The area of a triangle is equal to 1/2bh, where b is the base of the triangle and h is the height of the triangle. EOF is a right triangle, and its area is equal to 1/2 x 8v2 x 2v8 = 1/2 x 64 x 2 = 64 cm2. The size of the shaded area is equal to the area of EOF minus onefourth of the area of ABCD: 64 – 1/4 x 100 = 64 – 25 = 39 cm2 .
|
Download Printable PDF. VALID exam to help you PASS.
|
 |
