Final answer:
In triangle ABC, angle C is a right angle. Using the Pythagorean theorem and the given value of tan(A), we can find the lengths of sides AB and AC. AB is approximately 3.6 cm and AC is approximately 4.8 cm.
Step-by-step explanation:
In triangle ABC, angle C is a right angle, which means that triangle ABC is a right triangle.
The Pythagorean theorem can be used to find the lengths of the other two sides.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs (sides adjacent to the right angle) is equal to the square of the length of the hypotenuse (side opposite the right angle).
Let's assume that side AB is the opposite side (AC becomes the adjacent side) and side AC is the adjacent side (AB becomes the opposite side).
From the given information, tan(A) = 3/4.
Tangent is defined as the ratio of the opposite side (AB) to the adjacent side (AC) in a right triangle. So, we can set up the equation tan(A) = opposite/adjacent = AB/AC = 3/4.
Since angle C is a right angle, we can use the Pythagorean theorem to solve for AB and AC.
The theorem states that a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse. In this case, a = AB, b = AC, and c = BC.
We can substitute the given values and solve for AB and AC.
- Using tan(A) = opposite/adjacent = AB/AC = 3/4, we can say that AB = 3x and AC = 4x, where x is a common factor.
- From the Pythagorean theorem, we have a² + b² = c².
- Substituting the lengths, we get (AB)² + (AC)² = (BC)², which becomes (3x)² + (4x)² = 6².
- Simplifying the equation, we have 9x² + 16x² = 36.
- Combining like terms, we get 25x² = 36.
- Dividing both sides by 25, we find x² = 36/25.
- Taking the square root of both sides, we get x = √(36/25) = 6/5.
- Now, substituting the value of x, we can find AB and AC. AB = 3x = 3(6/5) = 18/5 = 3.6 cm and AC = 4x = 4(6/5) = 24/5 = 4.8 cm.