CAT — Polygons

Step 1: Assign coordinates to the hexagon. Let the side length of the regular hexagon be 2. Coordinates: A = (2, 0), B = (1, √3), C = (−1, √3), D = (−2, 0), E = (−1, −√3), F = (1, −√3). Since P and Q are midpoints: P = (3/2, √3/2), Q = (−3/2, √3/2).

Step 2: Find the lengths of the parallel sides of trapezium PBCQ. PQ = distance between P and Q = 3. BC = 2.

Step 3: Find the height of the trapezium. The y-coordinate of BC is √3. The y-coordinate of PQ is √3/2. Hence, height = √3 − √3/2 = √3/2.

Step 4: Find the area of trapezium PBCQ. Area = (1/2) × (sum of parallel sides) × height = (1/2) × (3 + 2) × (√3/2) = 5√3/4.

Step 5: Find the area of the hexagon. A regular hexagon consists of six equilateral triangles of side 2. Area of one equilateral triangle = (√3/4) × 2² = √3. Therefore, area of the hexagon = 6√3.

Step 6: Find the required ratio. Area of trapezium : Area of hexagon = (5√3/4) : 6√3 = 5 : 24.

Given:
A regular octagon has side length
6 cm.
Points A, C, E and G form a square.
Find the area of square ACEG.

Step 1: Find the circumradius of the octagon
For a regular octagon,
Side = 2R sin 22.5°
So,
6 = 2R sin 22.5°
Using,
sin 22.5° = √(2 − √2)/2,
6 = R√(2 − √2)
Therefore,
R = 6/√(2 − √2)

Step 2: Find the side of square ACEG
Vertices A and C are separated by a central angle of
90°.
Hence,
AC is the chord subtending 90°.
Therefore,
AC = R√2
Substituting the value of R,
AC = 6√2/√(2 − √2)
Square both sides,
AC² = 72/(2 − √2)
Rationalizing,
AC² = 72(2 + √2)/2
= 36(2 + √2)
Since AC is the side of the square,
Area of the square
= AC²
= 36(2 + √2)

Final Answer
Answer:
D. 36(2 + √2)