Answer:
1. The length of BC (the hypotenuse) using the Pythagorean theorem:
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as: a² + b² = c²
Here, AB = 12 cm and AC = 16 cm are the two sides of the triangle, and BC is the hypotenuse. Plugging in the given values:
(12 cm)² + (16 cm)² = BC²
144 cm² + 256 cm² = BC²
400 cm² = BC²
Taking the square root of both sides, we get:
BC = √400 cm² = 20 cm
2. Calculate the measure of angle B using trigonometric ratios:
We can use the tangent ratio, which is opposite side/adjacent side. In this case, the opposite side to angle B is AC and the adjacent side is AB. Therefore:
tan(B) = AC/AB = 16 cm / 12 cm = 4/3
To find angle B, we take the inverse tangent of 4/3:
B = tan⁻¹(4/3) ≈ 53.13°
3. Determine the area of the triangle ABC using the formula A = (1/2) * base * height:
In a right triangle, the two sides forming the right angle can be taken as the base and the height. Here, we can take AB as the base and AC as the height. Therefore:
A = 1/2 * AB * AC = 1/2 * 12 cm * 16 cm = 96 cm²
4. Find the length of the altitude from vertex A to side BC:
In a right-angled triangle, the altitude from the right angle vertex to the hypotenuse is the same as the side opposite the right angle. Therefore, the length of the altitude from vertex A to side BC is AC = 16 cm.