The angles of a triangle are in the ratio 2:3:5
1. Show that the triangle is right angled.
2. The length of the hypotenuse of the triangle is 12cm. Use trigonometry to calculate the length of the shortest side of this triangle
1. Let the angles of the triangle be 2x, 3x, and 5x. According to the given ratio, the angles are in the ratio 2:3:5, so we have:
2x + 3x + 5x = 180
10x = 180
x = 18
Therefore, the angles of the triangle are:
2x = 2(18) = 36 degrees
3x = 3(18) = 54 degrees
5x = 5(18) = 90 degrees
Since one of the angles is 90 degrees, the triangle is a right-angled triangle.
2. Let the sides of the right triangle be a, b, and c, with c being the hypotenuse. If c = 12 cm, then we can use trigonometry to calculate the length of the shortest side.
Using the Pythagorean theorem:
a^2 + b^2 = c^2
a^2 + b^2 = 12^2
a^2 + b^2 = 144
Since the sides are in the ratio 2:3:5, we can express the sides as:
a = 2k, b = 3k, c = 5k
Substitute these values into the Pythagorean theorem:
(2k)^2 + (3k)^2 = 144
4k^2 + 9k^2 = 144
13k^2 = 144
k^2 = 144 / 13
k^2 = 11.0769
k = √11.0769
k ≈ 3.3317
Therefore, the shortest side of the triangle is a:
a = 2k ≈ 2(3.3317) ≈ 6.6634 cm
So, the length of the shortest side of the triangle is approximately 6.6634 cm.