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AMC 10

Problem related to triangle – AMC 10B, 2019 Problem 10


The given problem is related to the calculation of area of triangle and distance between two points.

Try the problem


In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?

$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

2019 AMC 10B Problem 10

Problem related to triangle

6 out of 10

Secrets in Inequalities.

Knowledge Graph


Problem related to triangle- knowledge graph

Use some hints


Notice that it does not matter where the triangle is in the 2D plane so for our easy access we can select two points A and B in any place of choice.

So we can actually select any two points A and B such that they are 10 units apart so lets the points are \(A(0,0)\) and \(B(10,0)\) , as they are 10 units apart.

Now we can select the point C such that the perimeter of the triangle is 50 units. and then we can apply the formula of area to calculate the possible positions of C.

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AMC-8 USA Math Olympiad

Time and Distance Problem from AMC 8: Problem 6 – 2018

What are we learning ?

Competency in Focus: Time and Distance calculation

This problem from American Mathematics Contest 8 (AMC 8, 2018) is based on calculation of time and distance. It is Question no. 6 of the AMC 8 2018 Problem series.

First look at the knowledge graph:-

calculation of  mean and median- AMC 8 2013 Problem

Next understand the problem

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take? $\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100$
Source of the problem

American Mathematical Contest 2018, AMC 8 Problem 6

Key Competency

Basic Time and Distance problem with an easy interpretation from AMC 8 – 2018 – Problem 6

Difficulty Level
5/10
Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

Start with hints 

Do you really need a hint? Try it first!
Speed = \(\frac {distance}{time}\)  This can be the first hint for this sum. It is one of the important formula in science. Try to use it in this sum……..
So if we use the previous hint the speed would be  r = \(\frac {d}{t}\) so , r = \(\frac {10}{0.5}\)    r = 20 mph.
His speed on the highway then is $60$ mph. He drives $50$ miles, so he drives for $\frac{5}{6}$ hours, which is equal to $50$ minutes. Note : 60 miles\hour is equal to 1 mile\minute
I think you already got the answer but if not here is the last hint. The total amount of minutes spent on his trip is  =80 minutes

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