## Introduction

A triangle is a simple polygon with three sides and three angles. To solve problems involving triangles, it’s helpful to understand the components of a triangle. The base is the side of the triangle that is used as the reference for all other measurements. The height is the distance from the base to the opposite vertex. The area is the amount of space inside the triangle.

So why is it important to find the height of a triangle? For one thing, it’s often necessary to calculate the area of a triangle. The formula for the area of a triangle is A = 1/2bh, where b is the base and h is the height. Without knowing the height, you can’t accurately calculate the area. Additionally, being able to find the height of a triangle is a fundamental skill that can be applied to many other mathematical concepts.

## Method 1: Using the area formula

The most common way to find the height of a triangle is by using the formula for area. To refresh your memory, the formula for the area of a triangle is A = 1/2bh, where A is the area, b is the base, and h is the height.

If you know the area and the base of a triangle, you can isolate the height variable in the formula and solve for h. To do this, multiply both sides of the equation by 2 and divide by b:

A = 1/2bh

2A = bh

h = 2A/b

Let’s use an example to demonstrate this method:

Problem: Find the height of a triangle with a base of 6 cm and an area of 12 cm².

Solution:

Step 1: Write down the formula for the area of a triangle: A = 1/2bh.

Step 2: Plug in the known values: A = 12 cm² and b = 6 cm.

Step 3: Isolate the height variable in the formula: h = 2A/b.

Step 4: Plug in the values from Step 2 and solve for h: h = 2(12 cm²)/6 cm = 4 cm.

Therefore, the height of the triangle is 4 cm.

## Method 2: Using trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One way to find the height of a triangle using trigonometry is by using the tangent function. The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Here’s how you can use the tangent function to find the height of a triangle:

Step 1: Identify the right triangle that contains the height of the triangle you want to find. The height of the triangle is the side opposite the right angle.

Step 2: Label the sides of the triangle. The side opposite the angle you’re interested in is the opposite side. The side adjacent to the angle you’re interested in is the adjacent side. The hypotenuse is the side opposite the right angle.

Step 3: Write down the formula for tangent: tan(angle) = opposite/adjacent.

Step 4: Plug in the values you know and solve for the unknown. In this case, you want to find the height, which is the length of the opposite side.

Let’s use an example to demonstrate this method:

Problem: Find the height of a triangle with a base of 5 cm and an angle of 60°.

Solution:

Step 1: Draw a triangle with a base of 5 cm and label the angle as 60°.

Step 2: Identify the height of the triangle and label it as h.

Step 3: Write down the formula for tangent: tan(60°) = opposite/adjacent.

Step 4: Plug in the values you know and solve for the unknown: tan(60°) = h/5 cm.

Step 5: Simplify and solve for h: h = 5 cm x tan(60°) ≈ 8.66 cm.

Therefore, the height of the triangle is approximately 8.66 cm.

## Method 3: Using similar triangles

Another way to find the height of a triangle is by using similar triangles. Similar triangles are two triangles with the same shape but different sizes. In other words, the angles in one triangle are the same as the angles in the other triangle, but the lengths of the sides are different. When you have two similar triangles, you can set up a proportion and solve for the unknown variable.

Here’s how you can use similar triangles to find the height of a triangle:

Step 1: Draw a line from the vertex opposite the base to the base, creating two smaller triangles.

Step 2: Label the height of the larger triangle as h and the height of the smaller triangle as x.

Step 3: Label the base of the larger triangle as b and the base of the smaller triangle as a.

Step 4: Write down the formula for the similarity of triangles: a/b = x/h.

Step 5: Cross-multiply and solve for the unknown variable. In this case, you want to find the height, which is the length of h.

Let’s use an example to demonstrate this method:

Problem: Find the height of a triangle with a base of 8 cm and a height of 6 cm. A line is drawn from the vertex opposite the base to the base, creating two smaller triangles. One of the smaller triangles has a base of 4 cm.

Solution:

Step 1: Draw a diagram of the triangle and label the variables as described in the problem.

Step 2: Write down the formula for the similarity of triangles: a/b = x/h.

Step 3: Plug in the values you know and solve for the unknown: 4 cm/8 cm = x/6 cm.

Step 4: Cross-multiply and solve for x: 8x = 24 cm².

Step 5: Solve for x: x = 3 cm.

Step 6: Plug in the value of x to solve for h: 4 cm/8 cm = 3 cm/h.

Step 7: Cross-multiply and solve for h: 8h = 12 cm².

Step 8: Solve for h: h = 1.5 cm.

Therefore, the height of the triangle is 1.5 cm.

## Conclusion

There are three common methods for finding the height of a triangle: using the area formula, using trigonometry, and using similar triangles. By mastering these methods, you’ll be able to confidently tackle any triangle problem that comes your way. Remember to stay patient and practice regularly to improve your skills.