In triangle ABC, angle B is 90°. If cosec A = 13/12 and side AB = 10 cm, what is the length (in cm) of side AC?

Introduction / Context:
This question involves a right angled triangle and basic trigonometric ratios. You are told that angle B is 90 degrees, cosec A is 13/12 and one of the sides, AB, is 10 cm. You are asked to find the length of side AC. This tests understanding of right triangle geometry, the definition of cosecant and the Pythagoras theorem.

Given Data / Assumptions:

- Triangle ABC is right angled at B, so angle B = 90°. - Side AB = 10 cm. - cosec A = 13 / 12. - AC is the hypotenuse (opposite the right angle at B). - All sides are positive real lengths.

Concept / Approach:
In a right triangle with angle B = 90°, side AC is the hypotenuse, side BC is opposite angle A and side AB is opposite angle C. The trigonometric ratio sine of angle A is defined as opposite over hypotenuse, that is sin A = BC / AC. Its reciprocal, cosec A, is AC / BC. We use the given value of cosec A to express AC in terms of BC, then apply the Pythagoras theorem AC^2 = AB^2 + BC^2 to find the actual lengths. Finally we solve the resulting equations to determine AC.

Step-by-Step Solution:
Step 1: In triangle ABC, since angle B is 90°, AC is the hypotenuse. Step 2: For angle A, the side opposite A is BC and the hypotenuse is AC. Step 3: By definition, sin A = BC / AC and cosec A = 1 / sin A = AC / BC. Step 4: We are given that cosec A = 13 / 12, so AC / BC = 13 / 12. Step 5: This implies AC : BC = 13 : 12. We can write AC = 13k and BC = 12k for some positive k. Step 6: We also know AB = 10 cm, and Pythagoras theorem gives AC^2 = AB^2 + BC^2. Step 7: Substitute AC = 13k and BC = 12k into the Pythagoras formula: (13k)^2 = 10^2 + (12k)^2. Step 8: Compute the squares: 169k^2 = 100 + 144k^2. Step 9: Subtract 144k^2 from both sides: 169k^2 - 144k^2 = 25k^2 = 100. Step 10: Solve for k^2: k^2 = 100 / 25 = 4, so k = 2 (taking the positive root since k represents a length factor). Step 11: Now AC = 13k = 13 * 2 = 26 cm.

Verification / Alternative check:
Verify by finding BC as well. BC = 12k = 12 * 2 = 24 cm. Check Pythagoras: AB^2 + BC^2 = 10^2 + 24^2 = 100 + 576 = 676. The square root of 676 is 26, so AC = 26 cm, consistent with our result. Also, check cosec A: AC / BC = 26 / 24 = 13 / 12, matching the given value. Both checks confirm the correctness of AC = 26 cm.

Why Other Options Are Wrong:
An answer of 24 cm would incorrectly treat BC as the hypotenuse or misapply the trigonometric ratio. Values of 12 cm or 14 cm are too small to serve as the hypotenuse in a right triangle with a leg of length 10 cm and do not satisfy either the Pythagoras theorem or the given cosec A value. Only 26 cm is consistent with all the given information.

Common Pitfalls:
Some learners mix up which side is opposite angle A and which side is the hypotenuse, leading to a wrong ratio for sine and cosecant. Others may forget that angle B is the right angle and misidentify the hypotenuse. Carefully labeling the triangle and writing down the definitions of the trigonometric ratios before substituting values helps avoid these errors.

Final Answer:
The length of side AC is 26 cm, which corresponds to option D.