Pneminoultramicroscopic problem

Geometry Level 3

A triangle has altitudes of 12, 15 and 20 units. What is the largest angle (in degrees) in this triangle?

80 90 75 120

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Parth Lohomi by Parth Lohomi , 20, India

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4 solutions

Revanth Gumpu
Jul 6, 2014

I am pretty sure this approach is horrible, but its the way I solved it. Please feel free to comment. I had to know the relationship; altitude * side length = 2 * area I let x,y,z equal the three sides of the triangle and I set up equations;

12 * x = 2a <--- equation 1 15 * y = 2a <--- equation 2 20 * z = 2a <---equation 3

After, I divided each side by 2.

6x = a 7.5y = a 10z = a

Now I set equaton 2 = equation 1 and equation 3 = equation 1.

7.5 y = 6x AND 10z = 6x

I will solve for x/y and y/z respectively so I get; 7.5/6 = x/y AND 5/3 = x/z

Now I want to get a common number for x so I multiply the first proportion by 2 so I get 15 and I multiply the second equation by 3 to get 15 again.

Now I get 15/12 = x/y and 15/9.

As you can see now, x,y, and z are each 15,12,9 respectively. You can also see that 9-12-15 is a right triangle hence the answer being 90 degrees.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Nicely done. Note that there wasn't a need to find the exact side lengths of the triangle. As you pointed out, the ratio of sides is

1 12 : 1 15 : 1 20 = 5 : 4 : 3 \frac{1}{12} : \frac{1}{15} : \frac{1}{20} = 5 : 4 : 3

and hence we have a right triangle.

Calvin Lin Staff - 6 years, 11 months ago
Dollesin Joseph
Aug 21, 2014

I used cosine law. I assumed that the given 12,15,20 is the length of sides of triangle .

20² = 12² + 15² - 2(12)(15)cosC

400 = 144 + 225 - 360cosC

31 = -360cosC

-31/360 = cosC

arccos(-31/360) = C

C ≈ 94.939921132163051948018365682879

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Akash Deep
Aug 8, 2014

well i did it by using sine rule

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Thiago Virgínio
Jul 15, 2014

All triangles with 3, 4 and 5 as its sides are right triangles. 12, 15 and 20 is just this triangle, 4 times larger.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Note that you are given that the altitudes are 12, 15, 20, as opposed to the sides of the triangle.

A triangle with side lengths of 12, 15, 20 is not a right angled triangle.

Calvin Lin Staff - 6 years, 11 months ago

it can not be 90. If one angle is 90 then it has to follow pithagoras theoreum. So technically 12^2 + 15^2 should be 400 but instead its 369. So therefore it can't have a 90° angle. So the answer is wrong.

Saurabh Grover - 6 years, 11 months ago

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Altitudes, not sides

Josh Speckman - 6 years, 10 months ago

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