Chapter 2

Solve the given problems. Is it possible that the altitudes of a triangle meet, when extended, outside the triangle? Explain.

Describe the location of the midpoints of a set of parallel chords of a circle.

Solve the given problems. A walkway \(3.0 \mathrm{m}\) wide is constructed along the outside edge of a square courtyard. If the perimeter of the courtyard is \(320 \mathrm{m}\) (a) what is the perimeter of the square formed by the outer edge of the walkway? (b) What is the area of the walkway?

Solve the given problems. The altitude to the hypotenuse of a right triangle divides the triangle into two smaller triangles. What do you conclude about the original triangle and the two new triangles? Explain.

The measure of \(\widehat{A B}\) on a circle of radius \(r\) is \(45^{\circ} .\) What is the length of the arc in terms of \(r\) and \(\pi ?\)

Solve the given problems. An architect designs a rectangular window such that the width of the window is 18 in. less than the height. If the perimeter of the window is 180 in., what are its dimensions?

Solve the given problems. If two triangles have the same three angles, can you conclude that the triangles are congruent? Explain why or why not.

\(\text {Solve the given problems.}\) Spaceship Earth (shown in Fig. 2.127 ) at Epcot Center in Florida is a sphere of \(165 \mathrm{ft}\) in diameter. What is the volume of Spaceship Earth?

In a circle, a chord connects the ends of two perpendicular radii of 6.00 in. What is the area of the minor segment?

\(\text {Solve the given problems.}\) A special wedge in the shape of a regular pyramid has a square base \(16.0 \mathrm{mm}\) on a side. The height of the wedge is \(40.0 \mathrm{mm}\). What is the total surface area of the wedge (including the base)?

\(\text {Solve the given problems.}\) A lawn roller is a cylinder \(0.96 .0 \mathrm{m}\) long and \(0.60 \mathrm{m}\) in diameter. How many revolutions of the roller are needed to roll \(76 \mathrm{m}^{2}\) of lawn?

In \(1897,\) the Indiana House of Representatives passed unanimously a bill that included "the ... important fact that the ratio of the diameter and circumference is as five-fourths to four." Under this definition, what would be the value of \(\pi\) ? What is wrong with this House bill statement? (The bill also passed the Senate Committee and would have been enacted into law, except for the intervention of a Purdue professor.)

Solve the given problems. A \(1080 p\) high-definition widescreen television screen has 1080 pixels in the vertical direction and 1920 pixels in the horizontal direction. If the screen measures 15.8 in. high and 28.0 in. wide, find the number of pixels per square inch.

Solve the given problems. A perfect triangle is one that has sides that are integers and the perimeter and area are numerically equal integers. Is the triangle with sides \(6,25,\) and 29 a perfect triangle?

\(\text {Solve the given problems.}\) What is the area of a paper label that is to cover the lateral surface of a cylindrical can 3.00 in. in diameter and 4.25 in. high? The ends of the label will overlap 0.25 in. when the label is placed on the can.

Solve the given problems. Government guidelines require that a sidewalk to street ramp be such that there is no more than 1.0 in. rise for each horizontal 20.0 in. of the ramp. How long should a ramp be for a curb that is 4.0 in. above the street?

A person is in a plane \(11.5 \mathrm{km}\) above the shore of the Pacific Ocean. How far from the plane can the person see out on the Pacific? (The radius of Earth is \(6378 \mathrm{km}\).)

Solve the given problems. A fenced section of a ranch is in the shape of a quadrilateral whose sides are \(1.74 \mathrm{km}, 1.46 \mathrm{km}, 2.27 \mathrm{km},\) and \(1.86 \mathrm{km},\) the last two sides being perpendicular to each other. Find the area of the section.

Solve the given problems. The angle between the roof sections of an A-frame house is \(50^{\circ}\). What is the angle between either roof section and a horizontal rafter?

The CN Tower in Toronto has an observation deck at \(346 \mathrm{m}\) above the ground. Assuming ground level and Lake Ontario level are equal, how far can a person see from the deck? (The radius of Earth is 6378 km.)

Solve the given problems. A rectangular security area is enclosed on one side by a wall, and the other sides are fenced. The length of the wall is twice the width of the area. The total cost of building the wall and fence is 13,200 dollar. If the wall costs 50.00 dollar and the fence costs 5.00 dollar/m find the dimensions of the area.

Solve the given problems. A transmitting tower is supported by a wire that makes an angle of \(52^{\circ}\) with the level ground. What is the angle between the tower and the wire?

\(\text {Solve the given problems.}\) A ball bearing had worn down too much in a machine that was not operating properly. It remained spherical, but had lost \(8.0 \%\) of its volume. By what percent had the radius decreased?

An FM radio station emits a signal that is clear within 85 km of the transmitting tower. Can a clear signal be received at a home \(68 \mathrm{km}\) west and \(58 \mathrm{km}\) south of the tower?

Solve the given problems. What is the sum of the measures of the interior angles of a quadrilateral? Explain.

Solve the given problems. An 18.0 -ft tall tree is broken in a wind storm such that the top falls and hits the ground \(8.0 \mathrm{ft}\) from the base. If the two sections of the tree are still connected at the break, how far up the tree (to the nearest tenth of a foot) was the break?

A circular pool \(12.0 \mathrm{m}\) in diameter has a sitting ledge \(0.60 \mathrm{m}\) wide around it. What is the area of the ledge?

Solve the given problems. Find a formula for the area of a rhombus in terms of its diagonals \(d_{1}\) and \(d_{2} .\) (See Exercise 33.)

Solve the given problems. The Bermuda Triangle is sometimes defined as an equilateral triangle \(1600 \mathrm{km}\) on a side, with vertices in Bermuda, Puerto Rico, and the Florida coast. Assuming it is flat, what is its approximate area?

The radius of the Earth's equator is 3960 mi. What is the circumference?

Solve the given problems. The sail of a sailboat is in the shape of a right triangle with sides of \(8.0 \mathrm{ft}, 15 \mathrm{ft},\) and \(17 \mathrm{ft} .\) What is the area of the sail?

As a ball bearing rolls along a straight track, it makes 11.0 revolutions while traveling a distance of \(109 \mathrm{mm}\). Find its radius.

Solve the given problems. An observer is 550 m horizontally from the launch pad of a rocket. After the rocket has ascended \(750 \mathrm{m}\), how far is it from the observer?

The rim on a basketball hoop has an inside diameter of 18.0 in. The largest cross section of a basketball has a diameter of 12.0 in. What is the ratio of the cross-sectional area of the basketball to the area of the hoop?

Solve the given problems. In a practice fire mission, a ladder extended \(10.0 \mathrm{ft}\) just reaches the bottom of a 2.50 -ft high window if the foot of the ladder is \(6.00 \mathrm{ft}\) from the wall. To what length must the ladder be extended to reach the top of the window if the foot of the ladder is \(6.00 \mathrm{ft}\) from the wall and cannot be moved?

With no change in the speed of flow, by what factor should the diameter of a fire hose be increased in order to double the amount of water that flows through the fire hose?

Using a tape measure, the circumference of a tree is found to be 112 in. What is the diameter of the tree (assuming a circular cross section)?

Solve the given problems. A rectangular room is \(18 \mathrm{ft}\) long, \(12 \mathrm{ft}\) wide, and \(8.0 \mathrm{ft}\) high. What is the length of the longest diagonal from one corner to another corner of the room?

Solve the given problems. On a blueprint, a hallway is \(45.6 \mathrm{cm}\) long. The scale is \(1.2 \mathrm{cm}=1.0 \mathrm{m} .\) How long is the hallway?

Solve the given problems. The two sections of a folding door, hinged in the middle, are at right angles. If each section is \(2.5 \mathrm{ft}\) wide, how far are the hinges from the far edge of the other section?

Two pipes, each with a 25.0 -mm-diameter hole, lead into a single larger pipe (see Fig. 2.96). In order to ensure proper flow, the cross-sectional area of the hole of the larger pipe is designed to be equal to the sum of the cross- sectional areas of the two smaller pipes. Find the inside diameter of the larger pipe.

The velocity of an object moving in a circular path is directed tangent to the circle in which it is moving. A stone on a string moves in a vertical circle, and the string breaks after 5.5 revolutions. If the string was initially in a vertical position, in what direction does the stone move after the string breaks? Explain.

The cross section of a drainage trough has the shape of an isosceles triangle whose depth is \(12 \mathrm{cm}\) less than its width. If the depth is increased by \(16 \mathrm{cm}\) and the width remains the same, the area of the cross section is increased by \(160 \mathrm{cm}^{2}\). Find the original depth and width. See Fig. 2.50.