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problem solving involving triangle

problem solving involving triangle

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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.

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.

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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°

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

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 .

URL: https://www.purplemath.com/modules/rghtprob.htm

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problem solving involving triangle

Triangle Solving Practice

Practice solving triangles .

You only need to know:

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

Note: answers are rounded to 1 decimal place.

What Does "AAS", "ASA" etc Mean?

It means which sides or angles we already know:

Center for Problem oriented policing

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:

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:

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:

Check out the list of readings under the  POP Center recommended readings .

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Solving Oblique Triangles

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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

$ABC$

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."

$a_1, a_2, \ldots, a_n$

Introductory Problems

Intermediate Problems

Olympiad Problems

$a,b,c,d>0$

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Course: 8th grade   >   Unit 5

Triangle angle challenge problem

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Video transcript

IMAGES

  1. Question Video: Using Right Triangle Trigonometry to Solve Word Problems

    problem solving involving triangle

  2. Problem Solving

    problem solving involving triangle

  3. A triangle problem to solve

    problem solving involving triangle

  4. Solving Problems Involving Right Triangles

    problem solving involving triangle

  5. 😊 Solving right triangles word problems. Word Problems Using Right Triangles [Video]. 2019-01-30

    problem solving involving triangle

  6. Gr 10 Applied Math: Solving Problems Using Similar Triangles

    problem solving involving triangle

VIDEO

  1. Solving a Triangle

  2. 22-4 Solving Right Triangles

  3. Trigonometry: Trigonometric Ratios and Pythagoras Theorem

  4. Math problem involving angles in a triangle and trapezoid through adjacent polygons

  5. PROBLEM SOLVING INVOLVING HYPERBOLA

  6. PROBLEM SOLVING INVOLVING ELLIPSE

COMMENTS

  1. Triangles

    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

  2. Right triangle trigonometry word problems

    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.

  3. Solving Triangles

    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

  4. 3.4: Applications of Triangle Trigonometry

    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.

  5. Solving right-triangle word problems: Learn here!

    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.

  6. How to Solve Word Problems Involving Congruent Triangles

    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 &...

  7. Triangle Solving Practice

    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.

  8. The Problem Analysis 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.

  9. Solving Right Triangles: Problems

    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 .

  10. Solving Oblique Triangles: Problems

    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 .

  11. Triangle Inequality

    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 .

  12. Solving Word Problems Involving Similar Triangles

    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...

  13. Triangle angle challenge problem (video)

    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°.

  14. Congruent Triangles ( Real World )

    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 ...

  15. Solving Problems Involving Right Triangles

    @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...

  16. Law of Sines (Sine Law)

    @MathTeacherGon will demonstrate how to use the law of sines in solving problems in oblique triangles.The Six Trigonometric Ratios of Right Trianglehttps://...