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Chapter 5: Problem 1

Fill in the blanks. An ______ triangle is a triangle that has no right angle.

Short Answer

Expert verified
The triangle could be an acute or an obtuse triangle.

Step by step solution

01

Understand the Types of Triangles

According to the internal angles, triangles are classified into three types: (1) Acute triangle: All angles are less than 90 degrees. (2) Right triangle: One angle is exactly 90 degrees. (3) Obtuse triangle: One angle is more than 90 degrees. Since the exercise is looking for a triangle with no right angle, it could either be an acute or obtuse triangle.
02

Fill in the blank

Given that the triangle doesn't have any right angles, the options are only acute and obtuse triangle. However, the exercise doesn't provide enough information to conclude whether the triangle is acute or obtuse. Use both terms as possible answers, depending on any context cues.

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Key Concepts

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

acute triangle
An acute triangle is a special type of triangle where all of its interior angles are less than 90 degrees. This means no angle inside the triangle is a right angle or an obtuse angle. Think of the cute little angles that are sharp and pointed. In an acute triangle, each angle is squeezed tightly, making it rather narrow and slender.

This type of triangle can appear in various sizes and forms, but the sum of its angles will always total to 180 degrees, just like any other triangle. A key characteristic of acute triangles is that:
  • All angles measure less than 90 degrees.
  • It is impossible for an acute triangle to have a right or obtuse angle.
  • Your triangle will likely look very pointed or steep.
Knowing these attributes can help you spot an acute triangle whenever you're tackling geometry problems.
obtuse triangle
An obtuse triangle is where geometry takes a broader approach. This type of triangle has one angle that measures more than 90 degrees, known as the obtuse angle. Because of this larger angle, the obtuse triangle tends to look more spread out and wider compared to an acute triangle.

Most people find it easier to remember an obtuse angle as that one big and wide angle in a triangle, which tends to change the shape dynamically. Let's summarize its characteristics:
  • Exactly one angle is greater than 90 degrees.
  • The other two angles are, of course, less than 90 degrees.
  • Only one obtuse angle is possible in a triangle since having more would make it impossible for all angles to sum to 180 degrees.
Whenever you see a triangle with an angle that looks like it's stretching out more than a right angle, it's likely an obtuse triangle.
right triangle
The right triangle is perhaps the most famous triangle due to its characteristic right angle. This angle is exactly 90 degrees, making the triangle appear as though it has a perfect corner. Right triangles are essential in many areas of mathematics, particularly in trigonometry.

Here are some core features of right triangles:
  • Contains one right angle (exactly 90 degrees).
  • The longest side is called the hypotenuse, which is opposite the right angle.
  • The other two sides are referred to as the legs.
The scenarios where you typically see right triangles include building geometry, where precise angles are crucial, and in calculations where the Pythagorean theorem is applied.

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Most popular questions from this chapter

Verify the identity. $$a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \cos (B \theta-C)\( where \)C=\arctan (a / b)\( and \)b>0$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$7 \pi / 12$$

Prove the identity. $$\tan \left(\frac{\pi}{4}-\theta\right)=\frac{1-\tan \theta}{1+\tan \theta}$$

Use the formulas given in Exercises 89 and 90 to write the trigonometric expression in the form \(a \sin B \theta+b \cos B \theta\). $$5 \cos \left(\theta-\frac{\pi}{4}\right)$$

Find the exact value of the trigonometric expression given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\cot (v-u)$$
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