Careful observation

Geometry Level 3

In a triangle ABC with sides are 18, 24, 30, the distance between circumcentre and orthocentre of this triangle is $3a$ . Find $a$ .

The answer is 5.

2 solutions

Rishabh Jain
Jan 31, 2016

18, 24 and 30 form a Pythagorean triplet, hence given triangle is a right angled one and distance between orthocentre and circumcentre is simply circumradius of the triangle i.e half of hypotenuse i.e $\Large\boxed{\frac{30}{2}=15}$

Yes Same Way.

Kushagra Sahni - 5 years, 4 months ago

I didn't get it.Can you please elaborate?

Tanvir Hasan - 5 years, 4 months ago

need to explain this..

Safayet Ullah Neyam - 5 years, 4 months ago

$30^2=24^2+18^2$ .. Hence the triangle which would be formed would be a right angled one in which side of length 30 will form the hypotenuse. I have also added a figure in my solution for better visualisation .. Hope that helps..

Rishabh Jain - 5 years, 4 months ago

From diagram it is clear.Point A(vertex containing right angle) is orthocenter and mind point of hypotenuse is the circumcenter and its distance from A is half of the length of hypotenuse.

Deepak Kumar - 5 years, 4 months ago

Point A is orthocenter.

Deepak Kumar - 5 years, 4 months ago

Just a typo... Corrected :)

Rishabh Jain - 5 years, 4 months ago

Atomsky Jahid
Jan 31, 2016

I didn't observe it carefully. So, I had to devise coordinate axes. The vertices were (0, 0), (30, 0) and (10.8, 14.4). Then, I calculated the coordinates of orthocenter and circumcenter which are (10.8, 14.4) and (15, 0) respectively. After that I measured the distance between these two points which came out to be 15. Hence, the answer is 5.

Careful observation

The answer is 5.

2 solutions

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