All related to triangles Part 2

Geometry Level 2

Find the area of a triangle with sides 13 , 61 , 80 \sqrt{13}, \sqrt{61} , \sqrt{80} .


The answer is 14.

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5 solutions

Rohit Ner
May 25, 2015

There you go!

Moderator note:

Good. It would better to clarify that 13 \sqrt{13} is the hypotenuse of right triangle with sides 2 and 3. This goes for the other sides.

Bonus question: Using the same approach, find the area of a triangle with sides 29 , 85 , 146 \sqrt{29} , \sqrt{85}, \sqrt{146} .

Reply to Challenge Master:

We can imagine the following figure where the triangle's sides are the hypotenuse of other 3 triangles:

Then, the triangle area is:

77 5 21 27.5 = 23.5 77-5-21-27.5=\boxed{23.5}

Isaac Arce Aguilar - 6 years ago

Reply to Challenge Master: 77 square units if I'm not mistaken. I used pythagorean theorem btw :) and looked for what will fit with everything :3

Emmanuel David - 6 years ago

Reply to Challenge Master:

I get the answer as:

frac{7\sqrt{{{179}}{4}}

which is around 23.413 (3dp)

Syed Hamza Khalid - 4 years, 1 month ago
Chris Cheong
Apr 29, 2016

We can use the alternative form of Heron's formula:

A = 1 4 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) ( a 4 + b 4 + c 4 ) A=\frac { 1 }{ 4 } \sqrt { 2(a^{ 2 }b^{ 2 }+a^{ 2 }c^{ 2 }+b^{ 2 }c^{ 2 })-(a^{ 4 }+b^{ 4 }+c^{ 4 }) }

By squaring the side lengths, the square roots will be gone and the answer 14 can be easily obtained through some calculator spamming.

Manish Mayank
May 27, 2015

I solved it by using the formula 1 2 a b s i n C \frac{1}{2}ab sin C for the area of triangle and calculated s i n C sin C by using cosine law.

Amed Lolo
Jul 28, 2016

assume the Angle between two sides √13&√80 is P so cos(P)=(√80^2+√13^2-√61^2)(2×√80×√13)=4÷√65 so sin(P)=7÷√65 .area of∆=0.5×√80×√13×(7÷√65)=14#####

James Munday
May 27, 2015

This one was the exact same Same as the last one. When trying to solve the last one , instead of starting of by finding the areas of those smaller outside triangles, I used Pythagoras to find each f those outside triangles's hypotenuses. These were route 13 route 61 and route 80 so as soon as I saw this I knew the answer would be the same.

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