1st Edition

ISBN: 9780618800766

Author: Ron Larson, Lee Stiff, Laurie Boswell, Timothy Kanold

Publisher: Houghton Mifflin-High School

Concept explainers

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

Trigonometric Ratios

Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.

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Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 10.1, Problem 28E

To determine

To find whether or not the angle measure and side lengths of a triangle can have the same ratio of 3:4:5 and whose side lengths are in the same ratio.

Expert Solution & Answer

Answer to Problem 28E

It is not possible to have a triangle whose angle measures are in the ratio 3:4:5 and whose side lengths are in the same ratio.

Explanation of Solution

Given information:

Ratio of the angle measures in a triangle is given as 3:4:5

Calculation:

According to the sine rule, in a triangle with angles A, B, C and side lengths a, b and c

aSinA=bSinB=cSinC

Above can be rearranged as

a:b=SinA:SinB

And

b:c=SinB:SinC

Hence ratio of the side lengths are proportionate to the Sine of angles opposite to the side and hence cannot be same ratio as the angles

For the given example: let us assume the angles are 3x, 4x and 5x.

3x+4x+5x=180 ,

Now, add all the x together.

12x=180

Now, divide both the sides with 12

12x12=18012

This gives the value of x=15

First angle is 3x ,multiply 3 with 15

3×15=45

Second angle is 4x, multiply 4 with 15

4×15=60

Third angle is 5x, multiply 5 with 15

5×15=75

So, the 3 angles are 45, 60 and 75

The side ratio is Sin(45):Sin(60):Sin(75) which is not same as 3:4:5

Conclusion- It is not possible to have a triangle whose angle measures are in the ratio 3:4:5 and whose side lengths are in the same ratio.

Chapter 10.1, Problem 27E

Chapter 10.1, Problem 29E

Chapter 10 Solutions

Pre-Algebra

Ch. 10.1 - Prob. 1C Ch. 10.1 - Prob. 2C Ch. 10.1 - Prob. 3C Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E

Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.2 - Prob. 1C Ch. 10.2 - Prob. 2C Ch. 10.2 - Prob. 3C Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Prob. 15E Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - 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To find whether or not the angle measure and side lengths of a triangle can have the same ratio of 3 : 4 : 5 and whose side lengths are in the same ratio.