Answer:
a) the largest angle is 90°, making it a right triangle
b) the shortest side is 9.5 cm long
Step-by-step explanation:
You want the largest angle and the shortest side in a triangle whose angles are in the ratio 1 : 2 : 3.
Let x represent the smallest angle. Then the other two angles are 2x and 3x. Their sum is ...
x +2x +3x = 180°
x = 30° . . . . . divide by 6
3x = 90° . . . . find the largest angle
The largest angle is 90°, a right angle. So, the triangle is a right-angled triangle.
A triangle with angles of 30°, 60°, and 90° is a "special" right triangle. Its sides are in the ratio 1 : √3 : 2. That is, the shortest side is 1/2 the length of the longest side.
shortest side = 1/2(19 cm) = 9.5 cm
The length of the shortest side in the triangle is 9.5 cm.
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Additional comment
In case you're not familiar with the side length ratios of the 30-60-90 special triangle, you can figure the side length from the Law of Sines. That tells you ...
a/sin(A) = b/sin(B) = c/sin(C)
where A, B, C are the angles; and a, b, c are their opposite sides.
The shortest side is opposite the smallest angle, so we have ...
a/sin(30°) = (19 cm)/sin(90°)
a = sin(30°)×(19 cm) = 1/2(19 cm) = 9.5 cm
The length of the shortest side is 9.5 cm.