Finding the Missing Height of Acute Triangles

Finding the Missing Height of an Acute Triangle

• An acute triangle has all three angles less than 90 degrees.
• To find the height of an acute triangle, you need to know the base and area, or the base and the corresponding altitude.
• The altitude (height) of a triangle is a perpendicular line segment from a vertex to the opposite side (or an extension of the side).
• The area of a triangle is calculated by the formula: Area = (1/2) * base * height.

Explanation of the Method

• Given base and area: If you know the base and the area of the triangle, you can rearrange the formula to solve for the height.
  • Example: If the area is 24 square units and the base is 8 units, then height = (Area * 2) / base = (24 * 2) / 8 = 6 units.
• Given base and a side's length: If you know the base and length of one of the sides, you can use trigonometry (specifically, sine).
  • Draw an altitude from the vertex to the base, creating two right-angled triangles.
  • Determine the angle between the base and the side (often an exterior angle) using the triangle's other known angles.
  • Use the sine function: sin(angle) = (height) / (side length) .
  • Rearrange the equation to solve for the height: height = side length * sin(angle).

Summary

• Acute triangles, like others, have a predictable relationship between their base, height, and area.
• Understanding how to calculate the height is crucial for finding the triangle's area.
• Using the formula Area = (1/2) * base * height is fundamental.
• For cases where the height is not directly given, using trigonometry based on known side lengths and angles is a vital skill

Practice Problems

• Problem 1: An acute triangle has an area of 30 square centimeters and a base of 10 centimeters. Find the height.

• Solution 1: Height = (Area * 2) / base = (30 * 2) / 10 = 6 cm

• Problem 2: A triangle has a base of 12 units and one side of 15 units. The angle between the base and the 15-unit side measures 50 degrees. What is the height?

• Solution 2: Use the sine rule: height = 15 * sin(50°)≈ 11.5 units.

• Problem 3: Determine the altitude of an acute triangle with an area of 48 square feet and a base length of 12 feet.

• Solution 3: height = (Area * 2) / base = (48 * 2) / 12 = 8 feet

This quiz focuses on calculating the height of an acute triangle based on known parameters like base and area. It includes methods for finding the height using mathematical formulas and trigonometric principles, specifically for triangles with angles less than 90 degrees. Test your understanding of triangle properties and area calculations.