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Chapter 4: Problem 19

An equilateral triangle has sides of length \(8 \mathrm{~cm}\). a. Find the height of the triangle. (Hint: Use the Pythagorean theorem on the inside back cover.) b. Find the area \(A\) of the triangle if \(A=\frac{1}{2} b h\).

Short Answer

Expert verified
The height is approximately 6.93 cm, and the area is approximately 27.72 cm虏.

Step by step solution

01

Identify the properties of the equilateral triangle

An equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees. In this case, each side is given as 8 cm.
02

Divide the triangle into two right-angled triangles

Draw a height from one vertex to the midpoint of the opposite side. This splits the triangle into two right-angled triangles, each with legs of length 4 cm (half the base) and the height we need to find.
03

Apply the Pythagorean Theorem

In one of the right-angled triangles, use the Pythagorean Theorem to find the height. The formula is \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse (8 cm), \(a\) is half the base (4 cm), and \(b\) is the height. The equation is \(8^2 = 4^2 + h^2\).
04

Solve for the height

Rearrange the equation to solve for the height \(h\). \(64 = 16 + h^2 \ 64 - 16 = h^2 \ h^2 = 48 \ h = \sqrt{48} \ h = 4 \sqrt{3} \approx 6.93 \text{cm}\). So, the height \(h\) is approximately \(6.93\) cm.
05

Use the formula for the area of the triangle

The area formula for a triangle is \(A = \frac{1}{2} b h\). Here, the base \(b\) is 8 cm and the height \(h\) is approximately \(6.93\) cm.
06

Calculate the area

Substitute the values into the formula: \(A = \frac{1}{2} \times 8 \times 6.93 = 27.72 \text{cm}^2\). Thus, the area \(A\) is approximately \(27.72\) cm^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry. It is usually stated as follows: In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The formula is written as \(a^2 + b^2 = c^2\). Here, \(c\) is the hypotenuse, while \(a\) and \(b\) are the other two sides.
In the given exercise, we used the Pythagorean Theorem to find the height of an equilateral triangle. By splitting the equilateral triangle into two right-angled triangles, we consider one of these resulting right-angled triangles where the hypotenuse is 8 cm (the original side length) and one leg is 4 cm (half of the original side length). The other leg is the height we're looking for.
Applying the Pythagorean Theorem: \(8^2 = 4^2 + h^2\). Solving for \(h\), we find the height \(h\) of the equilateral triangle.
Height of a Triangle
The height of a triangle is a segment drawn from a vertex perpendicular to the opposite side (or the base). In the context of our exercise, the equilateral triangle's height was found using the Pythagorean Theorem.
Remember that in an equilateral triangle, drawing the height splits it into two congruent right-angled triangles. Each right-angled triangle has:
  • One leg as half the base: 4 cm.
  • The hypotenuse as the side of the equilateral triangle: 8 cm.
  • The other leg as the height: what we need to find.
Let's see why:
Using the equation we've derived from the Pythagorean Theorem, \(8^2 = 4^2 + h^2\), we solve it to get \(h = 4 \sqrt{3} \text{cm}\), approximately 6.93 cm. Thus, we found our height to proceed with other calculations.
Area of a Triangle
The area of a triangle is a measure of the space enclosed by the triangle. The standard formula to calculate the area is \(A = \frac{1}{2} \times base \times height\). In our problem, the base \((b)\) of the triangle is given as 8 cm, and the height \((h)\) was calculated to be approximately 6.93 cm.
Let's use this formula to determine the area:
  • Base (b) = 8 cm.
  • Height (h) = 6.93 cm.
Plug these values into the formula to get: \(A = \frac{1}{2} \times 8 \times 6.93 = 27.72 \text{cm}^2\). Hence, the area of this equilateral triangle is approximately 27.72 cm虏.
Understanding the area formula is crucial as it applies to any triangle - just need the base and height.

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