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Find m A D C in [link] . Round to the nearest tenth.

A triangle inside a triangle. The outer triangle is formed by vertices A, B, and D. Side B D is the base. The inner triangle shares vertices A and B. The last vertex C is located on the base side of the outer triangle between vertices B and D. Angle B is 60 degrees, side A D is 10, and side A C is 9.

Find A D in [link] . Round to the nearest tenth.

A triangle inside a triangle. The outer triangle is formed by vertices A, B, and D. Side B D is the base. The inner triangle shares vertices A and B. The last vertex C is located on the base side of the outer triangle between vertices B and D. Angle B is 53 degrees, angle D is 44 degrees, side A B is 12, and side A C is 13.

A D   13.8

Solve both triangles in [link] . Round each answer to the nearest tenth.

Two triangles formed by intersecting lines A D and B C. They intersect at point E. The first triangle is formed from vertices A, B, and E while the second triangle is formed from vertices C, E, and D. Angle A is 48 degrees, side A B is 4.2, angle D is 48 degrees, and side C D is 2. Angle A E B is 46 degrees.

Find A B in the parallelogram shown in [link] .

A parallelogram with vertices A, B, C, and D. There is a diagonal from vertex B to vertex C. Angle A is 130 degrees, angle D is 130 degrees, side B D is 10, and the diagonal B C is 12.

A B 2.8  

Solve the triangle in [link] . (Hint: Draw a perpendicular from H to J K ). Round each answer to the nearest tenth.

A triangle with vertices J, K, and H. Side J K is the horizontal base and is 10. Side JH is 7. Angle J is 20 degrees.

Solve the triangle in [link] . (Hint: Draw a perpendicular from N to L M ). Round each answer to the nearest tenth.

A triangle with vertices M, N, and L. Side M N is the horizontal base and is 4.6. Angle M is 74 degrees, and side M L is 5.

L 49.7 ,   N 56.3 ,   L N 5.8

In [link] , A B C D is not a parallelogram. m is obtuse. Solve both triangles. Round each answer to the nearest tenth.

A quadrilateral with vertices A, B, C, and D. There is a diagonal from vertex B to vertex D of length 45. Side A B is x, side B C is y, side C D is 40, and side D A is 29. Angle A is m degrees, angle C is 65 degrees, angle A B D is 35 degrees, angle D B C is n degrees, angle B D C is k degrees, and angle A D B is h degrees.

Real-world applications

A pole leans away from the sun at an angle of to the vertical, as shown in [link] . When the elevation of the sun is 55° , the pole casts a shadow 42 feet long on the level ground. How long is the pole? Round the answer to the nearest tenth.

A triangle within a triangle. The outer triangle is formed by vertices A, B, and S (the sun). Side A B is the horizontal base, the ground, and is 42 feet. Angle A is 55 degrees. The inner triangle is formed by vertices A, B, and C. Side B C is the pole. Vertex C is located on side A S of the outer triangle between vertices A and S. Angle C B S is 7 degrees.

51.4 feet

To determine how far a boat is from shore, two radar stations 500 feet apart find the angles out to the boat, as shown in [link] . Determine the distance of the boat from station A and the distance of the boat from shore. Round your answers to the nearest whole foot.

A triangle formed by the two radar stations A and B and the boat. Side A B is the horizontal base. Angle A is 70 degrees and angle B is 60 degrees.

[link] shows a satellite orbiting Earth. The satellite passes directly over two tracking stations A and B , which are 69 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 86.2° and 83.9° , respectively. How far is the satellite from station A and how high is the satellite above the ground? Round answers to the nearest whole mile.

A triangle formed by two ground tracking stations A and B and the satellite. Side A B is the horizontal base of the triangle. Angle A is 83.9 degrees, and the supplementary angle to angle B is 86.2 degrees.

The distance from the satellite to station A is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.

A communications tower is located at the top of a steep hill, as shown in [link] . The angle of inclination of the hill is 67° . A guy wire is to be attached to the top of the tower and to the ground, 165 meters downhill from the base of the tower. The angle formed by the guy wire and the hill is 16°. Find the length of the cable required for the guy wire to the nearest whole meter.

A triangle formed by the bottom of the hill, the base of the tower at the top of the hill, and the top of the tower. The side between the bottom of the hill and the top of the tower is wire. The length of the side bertween the bottom of the hill and the bottom of the tower is 165 meters. The angle formed by the wire side and the bottom of the hill is 16 degrees. The angle between the hill and the horizontal ground is 67 degrees.

The roof of a house is at a 20° angle. An 8-foot solar panel is to be mounted on the roof and should be angled 38° relative to the horizontal for optimal results. (See [link] ). How long does the vertical support holding up the back of the panel need to be? Round to the nearest tenth.

A triangle whose sides are the solar panel, the roof which goes past the solar panel, and the vertical support for the panel. The solar panel side is 8 feet long. There are horizontal dotted lines at the bottom of the solar panel and the bottom of the roof. The angle between the solar panel and the horizontal is 38 degrees. The angle between the roof and the horizontal is 20 degrees.

2.6 ft

Similar to an angle of elevation, an angle of depression is the acute angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.6 km apart, to be 37° and 44° , as shown in [link] . Find the distance of the plane from point A to the nearest tenth of a kilometer.

A triangle formed by points A and B on the ground and a plane in the air between them. Side A B is the horizontal ground. There is a horizontal dotted line parallel to the ground going through the plane. The angle formed by the dotted horizontal, the plane, and point A is 37 degrees. The angle between the dotted horizontal, the plane, and point B is 44 degrees.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4.3 km apart, to be 32° and 56°, as shown in [link] . Find the distance of the plane from point A to the nearest tenth of a kilometer.

A triangle formed between the plane and two points on the ground, A and B. Side A B is the horizontal base. The plane is above and to the left of both A and B. Point B is to the right of point A. There is a dotted horizontal line going through the plane parallel to the ground. The angle formed between point B, the plane, and the dotted horizontal line is 32 degrees. The angle formed between point A, the plane, and the dotted horizontal line is 56 degrees.

5.6 km

In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 39°. They then move 300 feet closer to the building and find the angle of elevation to be 50°. Assuming that the street is level, estimate the height of the building to the nearest foot.

In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35°. They then move 250 feet closer to the building and find the angle of elevation to be 53°. Assuming that the street is level, estimate the height of the building to the nearest foot.

371 ft

Points A and B are on opposite sides of a lake. Point C is 97 meters from A . The measure of angle B A C is determined to be 101°, and the measure of angle A C B is determined to be 53°. What is the distance from A to B , rounded to the nearest whole meter?

A man and a woman standing 3 1 2 miles apart spot a hot air balloon at the same time. If the angle of elevation from the man to the balloon is 27°, and the angle of elevation from the woman to the balloon is 41°, find the altitude of the balloon to the nearest foot.

5936 ft

Two search teams spot a stranded climber on a mountain. The first search team is 0.5 miles from the second search team, and both teams are at an altitude of 1 mile. The angle of elevation from the first search team to the stranded climber is 15°. The angle of elevation from the second search team to the climber is 22°. What is the altitude of the climber? Round to the nearest tenth of a mile.

A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of 28°. A 6-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is 28°. If the man and woman are 20 feet apart, how far is the street light from the tip of the shadow of each person? Round the distance to the nearest tenth of a foot.

24.1 ft

Three cities, A , B , and C , are located so that city A is due east of city B . If city C is located 35° west of north from city B and is 100 miles from city A and 70 miles from city B , how far is city A from city B ? Round the distance to the nearest tenth of a mile.

Two streets meet at an 80° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet.

19,056 ft 2

Brian’s house is on a corner lot. Find the area of the front yard if the edges measure 40 and 56 feet, as shown in [link] .

A triangle with angle 135 degrees. The sides adjacent to that angle are 56 feet and 40 feet. The other side is the house, length unknown.

The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles, the distance from Puerto Rico to Bermuda is 980 miles, and the angle created by the two distances is 62°.

445,624 square miles

A yield sign measures 30 inches on all three sides. What is the area of the sign?

Naomi bought a modern dining table whose top is in the shape of a triangle. Find the area of the table top if two of the sides measure 4 feet and 4.5 feet, and the smaller angles measure 32° and 42°, as shown in [link] .

A triangle. One angle is 32 degrees with opposite side = 4. Another angle is 42 degrees with opposite side = 4.5.

8.65 ft 2

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Berger describes sociologists as concerned with
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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