It's a Triangle!

Geometry Level 3

Let a a , b b , and c c be the lengths of the sides of a triangle . If the circumradius of the triangle is 3, and a × b × c = 120 a \times b \times c = 120 , then what is the area of this triangle ?


The answer is 10.

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3 solutions

Using the sine rule, a = 6 sin A b = 6 sin B c = 6 sin C a=6\sin A \\b=6\sin B\\c=6\sin C

The area of the triangle is given by Area = 1 2 a b sin C = 1 2 a b c 6 = 1 12 a b c = 10 \mbox{Area}=\frac{1}{2}ab\sin C=\frac{1}{2}ab\cdot\frac{c}{6}=\frac{1}{12}abc=10

Nice solution Jerry!

Akeel Howell - 4 years, 12 months ago
Akeel Howell
Jun 18, 2016

The area of a triangle can be written as a b c 4 R \frac{abc}{4R} , where a b c abc is the product of the length of the sides of the triangle, and R R is the circumradius of the triangle. Therefore, the area of this triangle is: 120 4 × 3 \frac{120}{4 \times 3} = 120 12 = \frac{120}{12} = 10 = \boxed{10}

Very nice area identity! =D

Pi Han Goh - 4 years, 12 months ago

Exactly what I did

Syed Hamza Khalid - 4 years, 1 month ago
Alejandro Miguel
Jun 20, 2016

a × b × c = 120 c / sin (A) = 2r = 6 ---> c = 6sin (A) Hence : a × b × 6sin (A) = 120 ---> a×b×sin (A) = 20 [a×b×sin (A)] / 2 = 10 ---> the area of a triangle

Sorry for not making it look fancy :P still learning.

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