Question

If the base of a triangle is 6 inches and its area is 60 square inches, what is the height of the triangle? (A) 5 inches (B) 10 inches (C) 20 inches (D) 180 inches

Short answer

The height of the triangle is \(20 \text{ inches}\) (C).

Step-by-step solution

Write down the given values

In this problem, the base of the triangle is 6 inches, and its area is 60 square inches. Let's denote the base as b and the area as A. So we have b = 6 inches and A = 60 square inches.

Write down the formula for the area of a triangle

The formula for the area of a triangle is given by: A = 1/2 × b × h, where h is the height of the triangle.

Plug in the given values

We have A = 60 square inches and b = 6 inches. Now, we can plug these values into the formula to get: 60 = 1/2 × 6 × h

Solve for the height h

Now, we will solve the equation to find the height h. We can first simplify the equation: 60 = 3 × h Next, we can divide both sides of the equation by 3: h = 60 ÷ 3 Now, we can find the value of h: h = 20 inches

Choose the correct answer from the options

Now, we have found that the height of the triangle is 20 inches. Looking at the given options, we can choose the correct answer: (C) 20 inches

Key concepts

Understanding the Area of a Triangle

The area of a triangle is a measure of the region enclosed by its three sides, given in square units. It can have different shapes based on the lengths of its sides and angles, but the formula to calculate the area remains consistent. The standard formula used to compute the area of a triangle is:

• \( A = \frac{1}{2} \times \text{base} \times \text{height} \)

Here, the base is one side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex. This formula works because a triangle can be thought of as half of a parallelogram, and therefore we divide the base times height by two to account for this halving.
This formula is derived from geometric principles and is critical in various applications, from architecture to computer graphics, because it allows us to quantify space in a meaningful way.

Steps in Calculating Triangle Height

The height of a triangle is a crucial part of solving many geometric problems and is often a variable you need to solve for using known values. To calculate the height of a triangle when you know the area and the base, you rearrange the triangle area formula to isolate the height:

• Start with the area formula: \( A = \frac{1}{2} \times b \times h \).
• Rearrange to solve for height: \( h = \frac{2A}{b} \).

In the original problem, we know:

• Area \( A = 60 \) square inches
• Base \( b = 6 \) inches

Plugging these into the rearranged formula gives us:

• \( h = \frac{2 \times 60}{6} \)
• \( h = \frac{120}{6} \)
• \( h = 20 \) inches

This calculation process demonstrates logical reasoning alongside mathematical manipulation to find unknown values.

Approaching Triangle Problems Strategically

Mathematics problem solving, particularly in triangle geometry, involves applying known formulas and logical reasoning. Solving such problems strategically requires understanding key relationships between the variables. First, identify the known values and what you need to find. Using formulas like the area of a triangle, rearrange to solve for unknowns:

• Identify the given quantities (e.g., base and area).
• Choose the correct formula for the solution.
• Rearrange the formula to solve for the required quantity (e.g., height).
• Perform the calculation step by step, avoiding skipping any steps to reduce error.

Through these steps, you ensure clarity and accuracy, which is crucial when solutions are used in real-world applications. Problem-solving in mathematics encourages analytical thinking, helping develop skills to tackle more complex mathematical challenges over time.