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Chapter 2: Problem 9

What is the difference between the numerical values of the area of the triangle and the length of the hypotenuse of the triangle? A. 1 B. 5 C. 6 D. 7

Short Answer

Expert verified
The difference between the numerical values of the area of the triangle and the length of the hypotenuse of the triangle is \(1\).

Step by step solution

01

Calculate the base and height from the given triangle

From the given triangle, we can see that the base and height have measures of 3 and 4, respectively. These will be used to calculate both the area and the hypotenuse length.
02

Calculate the area of the triangle

Using the area formula for a triangle: Area = (1/2) * base * height. Plugging in the base and height, we have: Area = (1/2) * 3 * 4 = (1/2) * 12 = 6
03

Calculate the hypotenuse length

Using the Pythagorean theorem: hypotenuse虏 = base虏 + height虏. Plugging in the base and height, we have: hypotenuse虏 = 3虏 + 4虏 = 9 + 16 = 25 Now, take the square root of both sides to find the length of the hypotenuse: hypotenuse = \(\sqrt{25}\) = 5
04

Calculate the difference between the area and the hypotenuse length

Now that we have the area of the triangle (6) and the length of the hypotenuse (5), we can find the difference: Difference = Area - Hypotenuse Length Difference = 6 - 5 = 1 The difference between the numerical values of the area of the triangle and the length of the hypotenuse of the triangle is 1. Therefore, the correct answer is: A. 1

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Key Concepts

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

Area Calculation
When trying to determine the area of a triangle, we can use a straightforward formula:
  • The area of a triangle is given by \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
This formula helps us to find out how much space is inside the triangle.
In our particular exercise, the base of the triangle is 3 and the height is 4. To find the area, simply plug these values into the formula: \[\text{Area} = \frac{1}{2} \times 3 \times 4 = 6.\]Thus, the area of this triangle is 6 square units, showing us the space inside our triangle.
Hypotenuse Length
To determine the hypotenuse length in a right-angled triangle, we utilize Pythagorean principles. The hypotenuse is the longest side of a right triangle, found opposite the right angle.
  • The Pythagorean theorem provides us with a way to calculate it easily.
In our exercise, the triangle has sides measuring 3 and 4.
Using Pythagorean theorem: \[hypotenuse^2 = base^2 + height^2\]Substituting our sides, we get: \[hypotenuse^2 = 3^2 + 4^2 = 9 + 16 = 25\]To find the hypotenuse length, take the square root of 25: \[hypotenuse = \sqrt{25} = 5.\]This calculation confirms that the hypotenuse, the longest side, is 5 units long.
Pythagorean Theorem
The Pythagorean theorem is an essential principle in triangle geometry, particularly for right-angled triangles.
  • It relates the squares of the lengths of the triangle's three sides.
Stating the theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This theorem is expressed in the formula: \[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
This theory is indispensable when calculating triangle properties and is frequently used in various mathematical and real-world applications.
For this specific triangle in our exercise, checking the side lengths verified the theorem holds true and allowed straightforward calculation of the hypotenuse.

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