A triangle has angles measuring 30°, 60°, and 90°. If the shortest side is 5 cm,

₹0.0

Question: A triangle has angles measuring 30°, 60°, and 90°. If the shortest side is 5 cm, what is the area?

Options:

Correct Answer: 12.5 cm²

Solution:

Area = 1/2 * base * height. Height = 5 * sin(60°) = 5 * (√3/2) = 5√3/2. Area = 1/2 * 5 * (5√3/2) = 12.5 cm².

A triangle has angles measuring 30°, 60°, and 90°. If the shortest side is 5 cm, what is the area?

A triangle has angles measuring 30°, 60°, and 90°. If the shortest side is 5 cm, what is the area?

Step 1: Identify the type of triangle. This triangle has angles of 30°, 60°, and 90°.
Step 2: Recognize that the shortest side is opposite the smallest angle (30°) and is given as 5 cm.
Step 3: Label the sides of the triangle. The side opposite the 30° angle (shortest side) is 5 cm.
Step 4: Use the properties of a 30-60-90 triangle. The side opposite the 60° angle is √3 times the shortest side: 5 * √3.
Step 5: The hypotenuse (opposite the 90° angle) is twice the shortest side: 2 * 5 = 10 cm.
Step 6: To find the area of the triangle, use the formula: Area = 1/2 * base * height.
Step 7: Choose the base as the shortest side (5 cm) and the height as the side opposite the 60° angle (5√3).
Step 8: Calculate the height: Height = 5 * sin(60°) = 5 * (√3/2) = 5√3/2.
Step 9: Substitute the base and height into the area formula: Area = 1/2 * 5 * (5√3/2).
Step 10: Simplify the area calculation: Area = 1/2 * 5 * (5√3/2) = 12.5 cm².

Triangle Properties – Understanding the properties of a 30-60-90 triangle, including the ratios of the sides.
Area Calculation – Using the formula for the area of a triangle and understanding the roles of base and height.
Trigonometric Functions – Applying sine to find the height of the triangle based on the angle measures.