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Chapter 1: Problem 1

If a triangle has a \(90^{\circ}\) angle, then it is a(n) ____triangle.

Short Answer

Expert verified
Right-angled

Step by step solution

01

Identify the Type of Triangle

A triangle with a 90-degree angle is classified based on its angles. Triangles are categorized as either acute, obtuse, or right-angled. A right-angled triangle has one 90-degree angle.
02

Definition of a Right Triangle

Triangles that have one angle exactly equal to 90 degrees are called right triangles. The right triangle is one of the fundamental types of triangles in geometry.

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Key Concepts

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

types of triangles
Triangles are one of the basic shapes in geometry. They have three sides and three angles. Understanding the different types of triangles can help in solving various problems in math and geometry.
Triangles are primarily classified based on their angles and their sides:
  • By Angles:
    • Acute Triangle: All three angles are less than 90 degrees.
    • Obtuse Triangle: One of the angles is greater than 90 degrees.
    • Right Triangle: One angle is exactly 90 degrees.

  • By Sides:
    • Equilateral Triangle: All three sides are of equal length, and all angles are 60 degrees.
    • Isosceles Triangle: Two sides are of equal length, and the angles opposite those sides are equal.
    • Scalene Triangle: All three sides and all three angles are different.
Knowing these types helps in identifying and solving problems related to triangles. In our specific exercise, we focused on the right triangle, which is crucial in many geometrical concepts and real-life applications.
geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. Triangles are a fundamental part of geometry.
Understanding triangles, especially right triangles, can help in solving many real-world problems. Right triangles are often used in construction, navigation, and even art.

In geometry, various theorems and formulas involve triangles:
  • Pythagorean Theorem: This relates to the sides of a right triangle, stated as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
  • Trigonometry: The study of relationships between the sides and angles of triangles. This includes sine, cosine, and tangent functions.
  • Area of a Triangle: Calculated using the formula \( A = \frac{1}{2} \times base \times height \).
Being comfortable with these concepts allows better understanding of more complex problems and designs.
Geometry, with its wide application, makes the study of triangles, especially right triangles, incredibly valuable.
90-degree angle
A 90-degree angle is one of the key elements in identifying right triangles. In geometric terms, a 90-degree angle is called a right angle. It is an angle that forms a perfect L shape.
Right triangles, by definition, have one 90-degree angle.

Some important points about 90-degree angles:
  • They are used to define perpendicular lines.
  • In a right triangle, the side opposite the 90-degree angle is the longest and is called the hypotenuse.
  • 90-degree angles are critical in construction for creating stable and square structures.
Understanding 90-degree angles is essential since they form the basis of trigonometry and many geometric concepts.
This angle helps in solving problems involving right triangles, making it a crucial aspect of geometric studies.

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Most popular questions from this chapter

A wheel is rotating at\(200 \mathrm{rev} / \mathrm{sec}\). Find the angular velocity in radians per minute (to the nearest tenth).

Solve each problem. If \(\tan \alpha=1 / 70,\) then what is \(\cot \alpha ?\)

Inscription Rock rises almost straight upward from the valley floor. From one point the angle of elevation of the top of the rock is \(16.7^{\circ} .\) From a point \(168 \mathrm{~m}\) closer to the rock, the angle of elevation of the top of the rock is \(24.1^{\circ} .\) How high is Inscription Rock? Round to the nearest tenth of a meter.

Find the length of the arc intercepted by the given central angle \(\alpha\) in a circle of radius \(r\). Round to the nearest tenth. $$ \alpha=3^{\circ}, r=4000 \mathrm{mi} $$

Use a calculator to find the value of each function. Round answers to four decimal places. $$ \sec \left(-48^{\circ} 3^{\prime} 12^{\prime \prime}\right) $$
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