Final answer:
To find the length of AB in right triangle ABC with a right angle at C, m²A = 55°, and CA = 10, one can use the Pythagorean theorem in conjunction with the tangent of angle A.
Step-by-step explanation:
In right triangle ABC, with m² C = 90°, m²A = 55°, and CA = 10, we are asked to find the length of AB to the nearest integer. According to the information given, we can use the Pythagorean theorem which relates the two legs of a right triangle with its hypotenuse as a² + b² = c². Given that CA (leg a or b) is 10 and angle C is 90°, the length of AB (hypotenuse c) can be found using the theorem.
Since we have one leg (CA) and the hypotenuse (AB) to find, we can rearrange the Pythagorean theorem to solve for AB: AB = √(CA² + CB²). But we do not have the length of CB, yet we can use the trigonometric functions to find CB since we know m²A and CA. Using the tangent of angle A (tan(55°) = CB/CA), we can find CB and then use the rearranged Pythagorean theorem to calculate the length of AB. Calculating the length of CB using the tangent function, and substituting the value into the Pythagorean theorem, will give us the length of AB, rounded to the nearest integer.