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Chapter 8: Problem 2

Tell whether a triangle with sides of the given lengths is acute, right, or obtuse. $$9,9,13$$

Short Answer

Expert verified
The triangle with sides 9, 9, 13 is an obtuse triangle.

Step by step solution

01

Identify the longest side

In our triangle the given sides are 9, 9, 13. The longest side (hypotenuse) of the triangle would be 13.
02

Square all the sides of the triangle

Square the lengths of all the sides, hence we obtain:\n\((9)^2 = 81\)\n\((9)^2 = 81\)\n\((13)^2 = 169\).
03

Compare the square of the longest side with sum of the squares of the other two sides

Add the squares of the lengths of the two shorter sides:\n\(81 + 81 = 162\).\nThen compare this with the square of the longest side (hypotenuse):\nIf \(169 > 162\) then the triangle is obtuse.

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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 triangle where all three interior angles are less than 90 degrees. This means each angle is sharp or acute, never reaching or exceeding a right angle. In terms of the sides, to check if a triangle is acute, you can use the Pythagorean theorem:
  • If the sum of the squares of the two shorter sides is greater than the square of the longest side, then the triangle is acute.
  • Mathematically, for sides \(a\), \(b\), and \(c\) (with \(c\) being the longest), the condition is: \(a^2 + b^2 > c^2\).
  • For example, if you had sides 5, 6, and 7, you would do \(5^2 + 6^2\) and compare it to \(7^2\). If the sum is greater, it is acute.
Acute triangles have some nice properties:
  • All altitudes (heights) fall within the interior of the triangle.
  • The circumcenter, which is the point where the perpendicular bisectors intersect, falls inside the triangle.
Understanding these characteristics helps identify and classify triangles effectively.
Right Triangle
A right triangle has one interior angle precisely equal to 90 degrees. The side opposite this right angle is called the hypotenuse, which is always the longest side. Right triangles are special and have some unique identities:
  • The Pythagorean theorem is fundamental here: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
  • This equation only holds exactly in right triangles.
  • Common examples are triangles with sides of lengths that are Pythagorean triples like \(3, 4, 5\) or \(5, 12, 13\).
Some interesting properties include:
  • The altitude from the right angle to the hypotenuse divides the triangle into two smaller, similar right triangles.
  • The centroid, orthocenter, and circumcenter are strategically positioned at known points in relation to the triangle.
Right triangles are used widely in real-world applications due to these unique and predictable properties.
Obtuse Triangle
An obtuse triangle has one interior angle that is greater than 90 degrees. This means it opens wider than a right angle. Here鈥檚 how to spot an obtuse triangle:
  • Use the Pythagorean theorem as an inequality: if the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse.
  • Mathematically, for sides \(a\), \(b\), and \(c\) (\(c\) being the longest): \(a^2 + b^2 < c^2\).
Obtuse triangles have some distinct features:
  • They appear to "lean" more, with one angle obviously larger than the others.
  • The orthocenter, which is the intersection of the altitudes, lies outside the triangle due to the angle size.
  • Only one angle, the obtuse one, is greater than 90 degrees; the others must be less than 90 degrees.
Understanding these helps in visually identifying and categorizing triangles in geometry.

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Most popular questions from this chapter

In \(\triangle A B C, m \angle B=m \angle C=72\) and \(B C=10\) a. Find \(A B\) and \(A C\) b. Find the length of the bisector of \(\angle A\) to \(\overline{B C}\).

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