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Right-Triangle Word Problems
What is a right-triangle word problem.
A right-triangle word problem is one in which you are given a situation (like measuring something's height) that can be modelled by a right triangle. You will draw the triangle, label it, and then solve it; finally, you interpret this solution within the context of the original exercise.
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Right Triangle Word Problems
Once you've learned about trigonometric ratios (and their inverses), you can solve triangles. Naturally, many of these triangles will be presented in the context of word problems. A good first step, after reading the entire exercise, is to draw a right triangle and try to figure out how to label it. Once you've got a helpful diagram, the math is usually pretty straightforward.
- A six-meter-long ladder leans against a building. If the ladder makes an angle of 60° with the ground, how far up the wall does the ladder reach? How far from the wall is the base of the ladder? Round your answers to two decimal places, as needed.
First, I'll draw a picture. It doesn't have to be good or to scale; it just needs to be clear enough that I can keep track of what I'm doing. My picture is:
To figure out how high up the wall the top of the ladder is, I need to find the height h of my triangle.
Since they've given me an angle measure and "opposite" and the hypotenuse for this angle, I'll use the sine ratio for finding the height:
sin(60°) = h/6
6 sin(60°) = h = 3sqrt[3]
Plugging this into my calculator, I get an approximate value of 5.196152423 , which I'll need to remember to round when I give my final answer.
For the base, I'll use the cosine ratio:
cos(60°) = b/6
6×cos(60°) = b = 3
Nice! The answer is a whole number; no radicals involved. I won't need to round this value when I give my final answer. Checking the original exercise, I see that the units are "meters", so I'll include this unit on my numerical answers:
ladder top height: about 5.20 m
ladder base distance: 3 m
Note: Unless you are told to give your answer in decimal form, or to round, or in some other way not to give an "exact" answer, you should probably assume that the "exact" form is what they're wanting. For instance, if they hadn't told me to round my numbers in the exercise above, my value for the height would have been the value with the radical.

- A five-meter-long ladder leans against a wall, with the top of the ladder being four meters above the ground. What is the approximate angle that the ladder makes with the ground? Round to the nearest whole degree.
As usual, I'll start with a picture, using "alpha" to stand for the base angle:
They've given me the "opposite" and the hypotenuse, and asked me for the angle value. For this, I'll need to use inverse trig ratios.
sin(α) = 4/5
m(α) = sin −1 (4/5) = 53.13010235...
(Remember that m(α) means "the measure of the angle α".)
So I've got a value for the measure of the base angle. Checking the original exercise, I see that I am supposed to round to the nearest whole degree, so my answer is:
base angle: 53°
- You use a transit to measure the angle of the sun in the sky; the sun fills 34' of arc. Assuming the sun is 92,919,800 miles away, find the diameter of the sun. Round your answer to the nearest whole mile.
First, I'll draw a picture, labelling the angle on the Earth as being 34 minutes, where minutes are one-sixtieth of a degree. My drawing is *not* to scale!:
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Hmm... This "ice-cream cone" picture doesn't give me much to work with, and there's no right triangle.
The two lines along the side of my triangle measure the lines of sight from Earth to the sides of the Sun. What if I add another line, being the direct line from Earth to the center of the Sun?
Now that I've got this added line, I have a right triangle — two right triangles, actually — but I only need one. I'll use the triangle on the right.
(The angle measure , "thirty-four arc minutes", is equal to 34/60 degrees. Dividing this in half is how I got 17/60 of a degree for the smaller angle.)
I need to find the width of the Sun. That width will be twice the base of one of the right triangles. With respect to my angle, they've given me the "adjacent" and have asked for the "opposite", so I'll use the tangent ratio:
tan(17/60°) = b/92919800
92919800×tan(17/60°) = b = 459501.4065...
This is just half the width; carrying the calculations in my calculator (to minimize round-off error), I get a value of 919002.8129 . This is higher than the actual diameter, which is closer to 864,900 miles, but this value will suffice for the purposes of this exercise.
diameter: about 919,003 miles
- A private plane flies 1.3 hours at 110 mph on a bearing of 40°. Then it turns and continues another 1.5 hours at the same speed, but on a bearing of 130°. At the end of this time, how far is the plane from its starting point? What is its bearing from that starting point? Round your answers to whole numbers.
The bearings tell me the angles from "due north", in a clockwise direction. Since 130 − 40 = 90 , these two bearings create a right angle where the plane turns. From the times and rates, I can find the distances travelled in each part of the trip:
1.3 × 110 = 143 1.5 × 110 = 165
Now that I have the lengths of the two legs, I can set up a triangle:
(The angle θ is the bearing, from the starting point, of the plane's location at the ending point of the exercise.)
I can find the distance between the starting and ending points by using the Pythagorean Theorem :
143 2 + 165 2 = c 2 20449 + 27225 = c 2 47674 = c 2 c = 218.3437657...
The 165 is opposite the unknown angle, and the 143 is adjacent, so I'll use the inverse of the tangent ratio to find the angle's measure:
165/143 = tan(θ)
tan −1 (165/143) = θ = 49.08561678...
But this angle measure is not the "bearing" for which they've asked me, because the bearing is the angle with respect to due north. To get the measure they're wanting, I need to add back in the original forty-degree angle:
distance: 218 miles
bearing: 89°
Related: Another major class of right-triangle word problems you will likely encounter is angles of elevation and declination .
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Triangle Solving Practice
Practice solving triangles .
You only need to know:
- Angles Add to 180°
- The Law of Sines
- The Law of Cosines
Try to solve each triangle yourself first, using pen and paper.
Then use the buttons to solve it step-by-step (more Instructions below).
Instructions
- Look at the triangle and decide whether you need to find another angle using 180°, or use the sine rule, or use the cosine rule. Click your choice.
- The formula you chose appears, now click on the variable you want to find.
- The calculation is done for you.
- Click again for other rules until you have solved the triangle.
Note: answers are rounded to 1 decimal place.
What Does "AAS", "ASA" etc Mean?
It means which sides or angles we already know:

- The Problem Analysis Triangle
While the SARA model is useful as a way of organizing the approach to recurring problems, it is often very difficult to figure out just exactly what the real problem is. The problem analysis triangle (sometimes referred to as the crime triangle) provides a way of thinking about recurring problems of crime and disorder. This idea assumes that crime or disorder results when (1) likely offenders and (2) suitable targets come together in (3) time and space, in the absence of capable guardians for that target. A simple version of a problem analysis triangle looks like this:
Offenders can sometimes be controlled by other people: those people are known as handlers. Targets and victims can sometimes be protected by other people as well: those people are known as guardians. And places are usually controlled by someone: those people are known as managers. Thus, effective problem-solving requires understanding how offenders and their targets/victims come together in places, and understanding how those offenders, targets/victims, and places are or are not effectively controlled. Understanding the weaknesses in the problem analysis triangle in the context of a particular problem will point the way to new interventions. A complete problem analysis triangle looks like this:
Problems can be understood and described in a variety of ways. No one way is definitive. They should be described in whichever way is most likely to lead to an improved understanding of the problem and effective interventions. Generally, incidents that the police handle cluster in four ways:
- Behavior . Certain behavior(s) is (are) common to the incidents. For example, making excessive noise, robbing people or businesses, driving under the influence, crashing vehicles, dealing drugs, stealing cars. There are many different behaviors that might constitute problems.
- Place . Certain places can be common to incidents. Incidents involving one or more problem behaviors may occur at, for example, a street corner, a house, a business, a park, a neighborhood, or a school. Some incidents occur in abstract places such as cyberspace, on the telephone, or through other information networks.
- Persons . Certain individuals or groups of people can be common to incidents. These people could be either offenders or victims. Incidents involving one or more behaviors, occurring in one or more places may be attributed to, for example, a youth gang, a lone person, a group of prostitutes, a group of chronic inebriates, or a property owner. Or incidents may be causing harm to, for example, residents of a neighborhood, senior citizens, young children, or a lone individual.
- Time . Certain times can be common to incidents. Incidents involving one or more behaviors, in one or more places, caused by or affecting one or more people may happen at, for example, traffic rush hour, bar closing time, the holiday shopping season, or during an annual festival.
There is growing evidence that, in fact, crime and disorder does cluster in these ways. It is not evenly distributed across time, place, or people. Increasingly, police and researchers are recognizing some of these clusters as:
- Repeat offenders attacking different targets at different places.
- Repeat victims repeatedly attacked by different offenders at different places.
- Repeat places (or hot spots) involving different offenders and different targets interacting at the same place.
The Problem Analysis Triangle was derived from the routine activity approach to explaining how and why crime occurs. This theory argues that when a crime occurs, three things happen at the same time and in the same space:
- a suitable target is available.
- there is the lack of a suitable guardian to prevent the crime from happening.
- a motivated offender is present.
Check out the list of readings under the POP Center recommended readings .
- What Is Problem-Oriented Policing?
- History of Problem-Oriented Policing
- Key Elements of POP
- The SARA Model
- Situational Crime Prevention
- 25 Techniques
- Links to Other POP Friendly Sites
- About POP en Español

Suggestions
- A Streetcar Named Desire
- The Crucible
- The Taming of the Shrew
- Twelfth Night
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Solving Oblique Triangles
- Study Guide
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Problem : Solve Triangle ABC given that: A = 45 o , B = 25 o , and a = 11 .
Problem : Solve Triangle ABC given that: A = 153 o , C = 15 o , and b = 11 .
Problem : Solve Triangle ABC given that: A = 35 o , B = 97 o , and C = 48 o .

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Triangle Inequality

That is, the sum of the lengths of any two sides is larger than the length of the third side. In degenerate triangles, the strict inequality must be replaced by "greater than or equal to."

- 1.1 Introductory Problems
- 1.2 Intermediate Problems
- 1.3 Olympiad Problems
Introductory Problems
- 2003 AMC 12A Problems/Problem 7
- 2006 AMC 10B Problem 10
- 2006 AIME II Problem 2
Intermediate Problems
- 2010 AMC 12A Problem 25
Olympiad Problems
- Belarus 2002 Aops Topic

- Geometric inequalities
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Course: 8th grade > Unit 5
- Angles in a triangle sum to 180° proof
- Isosceles & equilateral triangles problems
- Triangle exterior angle example
- Worked example: Triangle angles (intersecting lines)
- Worked example: Triangle angles (diagram)
Triangle angle challenge problem
- Triangle angle challenge problem 2
- Triangle angles review
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Video transcript

IMAGES
VIDEO
COMMENTS
You think they are useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined! Triangle types Learn Classifying triangles Classifying triangles by angles Worked example: Classifying triangles Types of triangles review Practice Classify triangles by angles 4 questions
Right triangle trigonometry word problems. CCSS.Math: HSG.SRT.C.8. Google Classroom. You might need: Calculator. Bugs Bunny was 33 33 meters below ground, digging his way toward Pismo Beach, when he realized he wanted to be above ground. He turned and dug through the dirt diagonally for 80 80 meters until he was above ground.
Six Different Types If you need to solve a triangle right now choose one of the six options below: Which Sides or Angles do you know already? (Click on the image or link) AAA Three Angles AAS Two Angles and a Side not between Two Angles and a Side between Two Sides and an Angle between Two Sides and an Angle not between Three Sides
We can now use the right triangle BDC to determine h as follows: h AC = sin(43.2 ∘) h = AC ⋅ sin(43.2 ∘) ≈ 59.980 So the top of the flagpole is 59.980 feet above the ground. This is the same answer we obtained in Exercise 3.4.1. Exercise 3.4.1 A bridge is to be built across a river.
Once you've learned about trigonometric ratios (and their inverses), you can solve triangles. Naturally, many of these triangles will be presented in the context of word problems. A good first step, after reading the entire exercise, is to draw a right triangle and try to figure out how to label it.
Step 1: Identify the givens. Step 2: Label the corresponding sides of the congruent triangles. Step 3: Use the data and side lengths of the triangles to solve the word problem. Equations &...
Instructions. Look at the triangle and decide whether you need to find another angle using 180°, or use the sine rule, or use the cosine rule. Click your choice. The formula you chose appears, now click on the variable you want to find. The calculation is done for you. Click again for other rules until you have solved the triangle.
The problem analysis triangle (sometimes referred to as the crime triangle) provides a way of thinking about recurring problems of crime and disorder. This idea assumes that crime or disorder results when (1) likely offenders and (2) suitable targets come together in (3) time and space, in the absence of capable guardians for that target.
Math Study Guide Topics Solving Right Triangles Right Triangle Review Techniques for Solving Problems Applications Problems Terms and Formulae Topics Problems Problem : Solve the following right triangle, in which C = 90o: a = 6, B = 40o . A = 90o - B = 50o. b = a tan (B) 5.0. c = 7.8 .
Math Study Guide Topics Formulae Topics Problems Problem : Solve Triangle ABC given that: A = 45o, B = 25o, and a = 11 . C = 180o - A - B = 110o. b = 6.6. c = 14.6 . Problem : Solve Triangle ABC given that: A = 153o, C = 15o, and b = 11 . B = 180o - A - C = 12o. a = 24.0. c = 13.7 .
The Triangle Inequality says that in a nondegenerate triangle: . That is, the sum of the lengths of any two sides is larger than the length of the third side. In degenerate triangles, the strict inequality must be replaced by "greater than or equal to.". The Triangle Inequality can also be extended to other polygons.The lengths can only be the sides of a nondegenerate -gon if for .
Step 1: Create a simple diagram to visually represent the similar triangles and the context of the problem. Step 2: Label the diagram with any side lengths given in the word problem and...
7 years ago. Imagine all of the lines that form the exterior angles extending outward to infinity. Now, imagine zooming out from the pentagon, until it shrinks to a point. You'll see all of the lines that we extended just converging to that point. Now, it's clear that all of those angles form a full circle, which is 360°.
How did the Egyptians build pyramids made up perfect congruent triangles? Search Bar. Search. Subjects. Explore. Donate. Sign In Sign Up. Click Create Assignment to assign this modality to your LMS. We have a new and improved read on this topic. ... Common Core Math; College FlexBooks; K-12 FlexBooks; Tools and Apps; v2.10.13.20230414011315 ...
@MathTeacherGon will solve problems involving right triangles. The main focus of this is to use trigonometric ratios in solving real life examples of right t...
@MathTeacherGon will demonstrate how to use the law of sines in solving problems in oblique triangles.The Six Trigonometric Ratios of Right Trianglehttps://...