How Cheenta works to ensure student success?

Explore the Back-StoryContents

[hide]

Try this beautiful problem from Geometry based on medians of triangle

In a triangle ABC, the medians from B to CA is perpendicular to the median from C to AB. If the median from A to BC is 30,determine \((BC^2 +AC^2+AB^2)/100\)?

- $56$
- $24$
- $34$

Geometry

Medians

Centroid

But try the problem first...

Answer:$24$

Source

Suggested Reading

PRMO-2018, Problem 10

Pre College Mathematics

First hint

We have to find out \((BC^2 +AC^2+AB^2)/100\). So, we have to find out \(AB, BC, CA\) at first. Now given that, the medians from B to CA is perpendicular to the median from C to AB and the median from A to BC is 30,

So clearly \(\triangle BGC\),\(\triangle BGF\),\(\triangle EGC\) are right angle triangle.Let \(CF=3x\) & \(BE =3y\) then clearly \(CG=2x\) & \(BG= 2y\) given that \(AD=30\) SO \(AG=20\) & \(DG =10\) (as \(G\) is centroid, medians intersects at 2:1). Therefore from pythagoras theorem we can find out \(BC,BF,CE\) i.e we can find out the value \(AB,BC,CA\)

Can you now finish the problem ..........

Second Hint

\(CE^2=(2x)^2+y^2=4x^2 +y^2\)

\(BF^2=(2y)^2+x^2=4y^2+x^2\)

Also, \(CG^2+BG^2=BC^2\) \(\Rightarrow 4x^2 + 4y^2={20}^2\) \(\Rightarrow x^2+y^2=100\)

\(AC^2=(2CE)^2=4(4x^2+y^2)\)

\(AB^2=(2BF)^2=4(4y^2+x^2)\)

Can you finish the problem........

Final Step

\((BC^2 +AC^2+AB^2)=20(x^2+y^2)+20^2=2400\)

so, \((BC^2 +AC^2+AB^2)/100\)=24

- https://www.cheenta.com/problem-on-semicircle-amc-8-2013-problem-20/
- https://www.youtube.com/watch?v=7AlfBAPWEMg

Contents

[hide]

Try this beautiful problem from Geometry based on medians of triangle

In a triangle ABC, the medians from B to CA is perpendicular to the median from C to AB. If the median from A to BC is 30,determine \((BC^2 +AC^2+AB^2)/100\)?

- $56$
- $24$
- $34$

Geometry

Medians

Centroid

But try the problem first...

Answer:$24$

Source

Suggested Reading

PRMO-2018, Problem 10

Pre College Mathematics

First hint

We have to find out \((BC^2 +AC^2+AB^2)/100\). So, we have to find out \(AB, BC, CA\) at first. Now given that, the medians from B to CA is perpendicular to the median from C to AB and the median from A to BC is 30,

So clearly \(\triangle BGC\),\(\triangle BGF\),\(\triangle EGC\) are right angle triangle.Let \(CF=3x\) & \(BE =3y\) then clearly \(CG=2x\) & \(BG= 2y\) given that \(AD=30\) SO \(AG=20\) & \(DG =10\) (as \(G\) is centroid, medians intersects at 2:1). Therefore from pythagoras theorem we can find out \(BC,BF,CE\) i.e we can find out the value \(AB,BC,CA\)

Can you now finish the problem ..........

Second Hint

\(CE^2=(2x)^2+y^2=4x^2 +y^2\)

\(BF^2=(2y)^2+x^2=4y^2+x^2\)

Also, \(CG^2+BG^2=BC^2\) \(\Rightarrow 4x^2 + 4y^2={20}^2\) \(\Rightarrow x^2+y^2=100\)

\(AC^2=(2CE)^2=4(4x^2+y^2)\)

\(AB^2=(2BF)^2=4(4y^2+x^2)\)

Can you finish the problem........

Final Step

\((BC^2 +AC^2+AB^2)=20(x^2+y^2)+20^2=2400\)

so, \((BC^2 +AC^2+AB^2)/100\)=24

- https://www.cheenta.com/problem-on-semicircle-amc-8-2013-problem-20/
- https://www.youtube.com/watch?v=7AlfBAPWEMg

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIALAcademic Programs

Free Resources

Why Cheenta?

How do you find that $BC = 20$ so that $BD = DC = 10.$

Oh! I got it. Because median of right triangle drawn from the vertex of right angle to the opposite side is half the length of it's hypotenuse.

yes...