ABCD is a trapezium where AB ⊥ BC ⊥ CD and ∠BAD is an obtuse angle. If tan ∠ADC = 0.75, CD = 7 cm and AD = 5 cm then the perimeter (in cm) of the trapezium ABCD is

To solve this problem, we need to find the lengths of all sides of the trapezium. We already know that CD = 7 cm and AD = 5 cm.

1. First, we need to find the length of DC. We know that tan ∠ADC = 0.75. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. So, in triangle ADC, tan ∠ADC = DC/AD = 0.75. Solving for DC, we get DC = AD * tan ∠ADC = 5 cm * 0.75 = 3.75 cm.

2. Next, we need to find the length of AB. Since AB is perpendicular to BC and BC is perpendicular to CD, triangle ABC is a right triangle. Therefore, we can use the Pythagorean theorem to find AB. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. So, AB^2 = BC^2 + AC^2. We know that BC = DC = 3.75 cm and AC = AD + CD = 5 cm + 7 cm = 12 cm. Substituting these values into the Pythagorean theorem, we get AB^2 = (3.75 cm)^2 + (12 cm)^2 = 14.0625 cm^2 + 144 cm^2 = 158.0625 cm^2. Taking the square root of both sides, we get AB = √158.0625 cm = 12.58 cm (approx).

3. Finally, we can find the perimeter of the trapezium by adding up the lengths of all its sides. The perimeter P = AB + BC + CD + AD = 12.58 cm + 3.75 cm + 7 cm + 5 cm = 28.33 cm.

So, the perimeter of the trapezium ABCD is approximately 28.33 cm.