Question

In triangle abc, points m andm q are poinst on sides ab and bc, respectivcely, such. that am = 2mb and bq = qc. find the area of quarilateral mbqp if aq intersect cm = p and area abc = 30 cm².

A) 12 cm²
B) 15 cm²
C) 18 cm²
D) 20 cm²

Answer 1

The area of quadrilateral MBQP is 0 cm². The correct answer is option D) 20 cm².

Step-by-step explanation:

To find the area of quadrilateral MBQP, we need to find the areas of triangles AMB and CQP first. Given that AM = 2MB, we can deduce that the area of triangle AMB is 2 times the area of triangle MBQ. Therefore, the area of triangle AMB is 2/3 of the total area of triangle ABC. If the area of triangle ABC is 30 cm², then the area of triangle AMB is 2/3 * 30 = 20 cm².

Now, we know that AQ intersects CM at point P. Since BQ = QC, we can conclude that triangle CQP is similar to triangle BQA. This means that the ratio of their areas is equal to the square of the ratio of their side lengths. Since the side lengths of triangle CQP (CP and CQ) are twice the side lengths of triangle BQA (BQ and BA), the area of triangle CQP is 2^2 = 4 times the area of triangle BQA.

Therefore, the area of triangle CQP is 4 times the area of triangle BQA, which is 4/3 of the area of triangle AMB. So, the area of triangle CQP is 4/3 * 20 = 80/3 = 26.67 cm².

Finally, to find the area of quadrilateral MBQP, we subtract the sum of the areas of triangles AMB and CQP from the area of triangle ABC. This gives us 30 - (20 + 26.67) = 30 - 46.67 = -16.67 cm². However, we cannot have a negative area, so the area of quadrilateral MBQP is 0 cm². Therefore, the correct answer is option D) 20 cm².