Chapter 2: Trigonometric Ratios
Exercises: 2.1 Side and Angle Relationships
Exercises Homework 2.1
Skills
- Identify inconsistencies in figures #1-12
- Use the triangle inequality to put bounds on the lengths of sides #13-16
- Use the Pythagorean theorem to find the sides of a right triangle #17-26
- Use the Pythagorean theorem to identify right triangles #27-32
- Solve problems using the Pythagorean theorem #33-42
Suggested homework problems
Exercise Group
For Problems 1–12, explain why the measurements shown cannot be accurate.
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13.
If two sides of a triangle are [latex]6[/latex] feet and [latex]10[/latex] feet long, what are the largest and smallest possible values for the length of the third side?
14.
Two adjacent sides of a parallelogram are [latex]3[/latex] cm and [latex]4[/latex] cm long. What are the largest and smallest possible values for the length of the diagonal?
15.
If one of the equal sides of an isosceles triangle is [latex]8[/latex] millimeters long, what are the largest and smallest possible values for the length of the base?
16.
The town of Madison is [latex]15[/latex] miles from Newton and [latex]20[/latex] miles from Lewis. What are the possible values for the distance from Lewis to Newton?
Exercise Group
For Problems 17–22,
- Make a sketch of the situation described and label a right triangle.
- Use the Pythagorean theorem to solve each problem.
17.
The size of a TV screen is the length of its diagonal. If the width of a [latex]35[/latex]-inch TV screen is [latex]28[/latex] inches, what is its height?
18.
If a [latex]30[/latex]-meter pine tree casts a shadow of [latex]30[/latex] meters, how far is the tip of the shadow from the top of the tree?
19.
The diagonal of a square is [latex]12[/latex] inches long. How long is the side of the square?
20.
The length of a rectangle is twice its width, and its diagonal is [latex]9\sqrt{5}[/latex] meters long. Find the dimensions of the rectangle.
21.
What size rectangle can be inscribed in a circle of radius [latex]30[/latex] feet if the length of the rectangle must be three times its width?
22.
What size square can be inscribed inside a circle of radius [latex]8[/latex] inches so that its vertices just touch the circle?
Exercise Group
For Problems 23–26, find the unknown side of the triangle.
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Exercise Group
For Problems 27–32, decide whether a triangle with the given sides is a right triangle.
27.
[latex]9[/latex] in, [latex]16[/latex] in, [latex]25[/latex] in
28.
[latex]12[/latex] m, [latex]16[/latex] m, [latex]20[/latex] m
29.
[latex]5[/latex] m, [latex]12[/latex] m, [latex]13[/latex] m
30.
[latex]5[/latex] ft, [latex]8[/latex] ft, [latex]13[/latex] ft
31.
[latex]5^2[/latex] ft, [latex]8^2[/latex] ft, [latex]13^2[/latex] ft
32.
[latex]\sqrt{5}[/latex] ft, [latex]\sqrt{8}[/latex] ft, [latex]\sqrt{13}[/latex] ft
33.
Show that the triangle with vertices [latex](0,0) \text{,} (6,0)[/latex] and [latex](3,3)[/latex] is an isosceles right triangle; that is, a right triangle with two sides of the same length.
34.
Two opposite vertices of a square are [latex]A(-9,-5)[/latex] and [latex]C(3,3) \text{.}[/latex]
- Find the length of a diagonal of the square.
- Find the length of the side of the square.
35.
A [latex]24[/latex]-foot flagpole is being raised by a rope and pulley, as shown in the figure. The loose end of the rope can be secured to a ring on the ground [latex]7[/latex] feet from the base of the pole. From the ring to the top of the pulley, how long should the rope be when the flagpole is vertical?
36.
To check whether the corners of a frame are square, carpenters sometimes measure the sides of a triangle, with two sides meeting at the join of the boards. Is the corner shown in the figure square?
Exercise Group
37.
Find [latex]\alpha, \beta[/latex], and [latex]h\text{.}[/latex]
38.
Find [latex]\alpha, \beta[/latex], and [latex]d\text{.}[/latex]
39.
Find the diagonal of a cube of side [latex]8[/latex] inches. Hint: Find the diagonal of the base first.
40.
Find the diagonal of a rectangular box whose sides are [latex]6[/latex] cm by [latex]8[/latex] cm by [latex]10[/latex] cm. Hint: Find the diagonal of the base first.
Exercise Group
For Problems 41 and 42, make a sketch and solve.
41.
- The back of Brian’s pickup truck is 5 feet wide and 7 feet long. He wants to bring home a [latex]9[/latex]-foot length of copper pipe. Will it lie flat on the floor of the truck?
- Find the length of the side of the square.
42.
What is the longest curtain rod that will fit inside a box [latex]60[/latex] inches long by [latex]10[/latex] inches wide by [latex]4[/latex] inches tall?
43.
In this problem, we’ll show that any angle inscribed in a semicircle must be a right angle. The figure shows a triangle inscribed in a unit circle, one side lying on the diameter of the circle and the opposite vertex at point [latex][latex]p,q[/latex][/latex] on the circle.
- What are the coordinates of the other two vertices of the triangle? What is the length of the side joining those vertices?
- Use the distance formula to compute the lengths of the other two sides of the triangle.
- Show that the sides of the triangle satisfy the Pythagorean theorem, [latex]a^2 + b^2 = c^2\text{.}[/latex]
44.
There are many proofs of the Pythagorean theorem. Here is a simple visual argument.
- What is the length of the side of the large square in the figure? Write an expression for its area.
- Write another expression for the area of the large square by adding the areas of the four right triangles and the smaller central square.
- Equate your two expressions for the area of the large square and deduce the Pythagorean theorem.