Problem 13 Determine the type of triangle r... [FREE SOLUTION]

Chapter 5: Problem 13

Determine the type of triangle represented if the lengths of its sides are: a) \(a=4, b=3,\) and \(c=5\) b) \(a=4, b=5,\) and \(c=6\) c) \(a=2, b=\sqrt{3},\) and \(c=\sqrt{7}\) d) \(a=3, b=8,\) and \(c=15\)

Short Answer

Expert verified

a) Right triangle, b) Scalene triangle, c) Scalene triangle, d) Not a triangle.

Step by step solution

Identify the Triangle Type with Sides a = 4, b = 3, c = 5

To determine the type of triangle, we use the Pythagorean theorem: a triangle is a right triangle if \( c^2 = a^2 + b^2 \). For sides \( a=4 \), \( b=3 \), and \( c=5 \), calculate: \( 5^2 = 25 \), \( 4^2 + 3^2 = 16 + 9 = 25 \). Since \( 25 = 25 \), the triangle is a right triangle.

Determine Triangle Type for Sides a = 4, b = 5, c = 6

Check whether any triangle inequality or Pythagorean theorem holds. First, the triangle inequality: any two sides' sum should be greater than the third side. For all combinations, \( 4+5 > 6 \), \( 5+6 > 4 \), \( 4+6 > 5 \) hold true. Hence, it forms a triangle. For the Pythagorean theorem: \( 6^2 = 36 \), \( 4^2 + 5^2 = 16 + 25 = 41 \). Since \( 41 eq 36 \), it's not a right triangle. Therefore, it is a scalene triangle.

Check Triangle Type with Sides a = 2, b = \( \sqrt{3} \), c = \( \sqrt{7} \)

Use the triangle inequality: \( a + b > c \), \( a + c > b \), \( b + c > a \). Calculate using approximate values \( \sqrt{3} \approx 1.73 \), \( \sqrt{7} \approx 2.65 \). Check inequalities: \( 2 + 1.73 > 2.65 \), \( 2 + 2.65 > 1.73 \), \( 1.73 + 2.65 > 2 \). All hold true, forming a triangle. Compare \( c^2 \approx 7 \) to other powers \( a^2 + b^2 \approx 2 + 3 = 5 \), no Pythagorean, so it is a scalene triangle.

Analyze Possible Triangle for Sides a = 3, b = 8, c = 15

Apply the triangle inequality: \( a + b \) must be greater than \( c \). Calculate \( 3 + 8 = 11 \), which is not greater than 15, failing the inequality test. Therefore, these sides cannot form a triangle.

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Key Concepts

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

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It applies specifically to right triangles. A triangle is considered to be a right triangle if the square of its longest side (the hypotenuse) is equal to the sum of the squares of the other two sides. Let's break it down further:

In a triangle with sides labeled as \( a \), \( b \), and \( c \) where \( c \) is the hypotenuse, the theorem states: \( c^2 = a^2 + b^2 \).
This relationship is only valid for right triangles, which have one 90-degree angle.
If \( c^2 = a^2 + b^2 \), the triangle is right.

If this equation holds true, then the triangle is a right triangle, just as seen in Example a), where \( a=4 \), \( b=3 \), and \( c=5 \). The measurements satisfy the theorem since \( 5^2 = 4^2 + 3^2 \), so the triangle is right-angled. Understanding this allows you to classify triangles as right triangles and understand their geometric properties better.

Triangle Inequality

The Triangle Inequality Theorem is central to determining if three given side lengths can form a triangle. This theorem asserts that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Consider the following:

For three side lengths \( a \), \( b \), and \( c \), the following must be true for a valid triangle to form:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
If even one of these inequalities doesn't hold, the side lengths cannot form a triangle.

For example, in Example d) with sides \( a=3 \), \( b=8 \), and \( c=15 \), the inequality \( a+b > c \) fails because \( 3 + 8 = 11 \) is not greater than 15. Thus, no triangle can form with these side lengths. Understanding the Triangle Inequality Theorem helps in quickly assessing the feasibility of forming a triangle with given sides.

Types of Triangles

Triangles can be classified into several types based on their sides and angles. Let's explore the basic types that are important for understanding geometrical properties:

Equilateral Triangle: This triangle has all three sides equal and all angles equal to 60 degrees.
Isosceles Triangle: This triangle has at least two sides that are equal. Its base angles are also equal.
Scalene Triangle: A triangle with all sides and angles different. Scalene triangles do not have any equal sides or angles.
Right Triangle: This triangle has one angle that is exactly 90 degrees.You can see examples of different triangle types in the step-by-step solution. Example b) with sides \( a=4 \), \( b=5 \), and \( c=6 \) is classified as a scalene triangle, as none of its sides are equal. Same with Example c), where approximate calculations showed that \( a=2 \), \( b=\sqrt{3} \), and \( c=\sqrt{7} \) also formed a scalene triangle. Knowledge of triangle types aids in visualizing and solving geometric problems involving triangles.

91影视

Determine the type of triangle represented if the lengths of its sides are: a) \(a=4, b=3,\) and \(c=5\) b) \(a=4, b=5,\) and \(c=6\) c) \(a=2, b=\sqrt{3},\) and \(c=\sqrt{7}\) d) \(a=3, b=8,\) and \(c=15\)

Short Answer

Step by step solution

Identify the Triangle Type with Sides a = 4, b = 3, c = 5

Determine Triangle Type for Sides a = 4, b = 5, c = 6

Check Triangle Type with Sides a = 2, b = \( \sqrt{3} \), c = \( \sqrt{7} \)

Analyze Possible Triangle for Sides a = 3, b = 8, c = 15

Key Concepts

Pythagorean Theorem

Triangle Inequality

Types of Triangles

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