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Chapter 11: Problem 2

Find the area of each figure. A triangle with sides \(3,4,\) and 5

Short Answer

Expert verified
The area of the triangle is 6 square units.

Step by step solution

01

Identify the base and the height

As the given triangle is a right triangle with sides measuring 3, 4, and 5, the sides measuring 3 and 4 can be considered as the base and the height of the triangle.
02

Use the formula for the area of a right triangle

The formula for the area of a triangle is given by \( \frac{1}{2} \times base \times height \). Plugging in the values, the base as 3 and height as 4, we get \( \frac{1}{2} \times 3 \times 4 \).
03

Calculate the area

Upon calculating \( \frac{1}{2} \times 3 \times 4 \), we find the area of the triangle to be 6 square units.

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Key Concepts

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

Right Triangle
A right triangle is a type of triangle that contains one angle equal to 90 degrees. This right angle is what sets it apart from other types of triangles. In our exercise, the triangle with sides measuring 3, 4, and 5 is a perfect example of a right triangle. The longest side, known as the hypotenuse, is opposite the right angle.
In a right triangle:
  • One angle is exactly 90 degrees, known as the right angle.
  • The side opposite the right angle is the hypotenuse.
  • The other two sides are called the legs.
Right triangles are interesting because they have unique mathematical relationships, such as those described by the Pythagorean theorem, which we'll dive into next.
Pythagorean Theorem
The Pythagorean theorem is a formula central to geometry, particularly when dealing with right triangles. It states that for a right 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. Mathematically, it's expressed as:
  • \[ a^2 + b^2 = c^2 \]
Where:
  • \(a\) and \(b\) are the lengths of the legs, and
  • \(c\) is the length of the hypotenuse.
In our example of the triangle with sides 3, 4, and 5, we can check that it is a right triangle by verifying this theorem:
  • \[ 3^2 + 4^2 = 9 + 16 = 25 \]
  • Indeed, \(5^2 = 25\), confirming the Pythagorean theorem holds true.

This brief exploration showcases the power and application of the Pythagorean theorem, not only in identifying right triangles but also in solving various geometric problems.
Geometric Formulas
In geometry, formulas are essential for calculating various properties of shapes. For triangles, one of the most important formulas is the area calculation.The area of a triangle can be found using the following formula:
  • \[ ext{Area} = rac{1}{2} imes ext{base} imes ext{height} \]
This formula applies seamlessly to right triangles. In a right triangle, one can conveniently choose the two legs as the base and the height. For our triangle with sides 3, 4, and 5:
  • The base is 3 and the height is 4.
  • Substituting into the area formula gives us:
  • \[ rac{1}{2} imes 3 imes 4 = 6 \]
Therefore, the area of this right triangle is 6 square units. This simple use of geometric formulas helps break down problems, making them easier to manage and solve.

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Most popular questions from this chapter

Find the area of each figure. A square inscribed in a circle with radius \(r\)

Use a calculator or the trigonometry table on page 311 to find the area of each figure to the nearest tenth. An isosceles triangle with a \(32^{\circ}\) vertex angle and a base of \(8 \mathrm{cm}\)

Draw a scalene triangle \(A B C\) Construct an equilateral triangle whose area is equal to the area of \(\triangle A B C\).

Draw a rectangle. Then construct a square with equal area.

A slide projector casts a circle of light with radius \(2 \mathrm{ft}\) on a screen that is 10 ft from the projector. If the screen is removed, the projector shines an even larger circle of light on the wall that was 10 ft behind the screen. Find the circumferences and areas of the circles of light on the screen and on the wall. Leave answers in terms of \(\pi\).
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