110k views
3 votes
The angles in a triangle are in the ratio 1 : 2 : 3.

a) Show that the triangle is a right-angled triangle.

b) The hypotenuse of the triangle is 19 cm long.

Calculate the length of the shortest side in the triangle.

User Amigolargo
by
7.3k points

1 Answer

0 votes

Answer:

a) the largest angle is 90°, making it a right triangle

b) the shortest side is 9.5 cm long

Explanation:

You want the largest angle and the shortest side in a triangle whose angles are in the ratio 1 : 2 : 3.

Angles

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.

Sides

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.

__

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.

User Richie Li
by
6.8k points