1964 AHSME Problems/Problem 35
Problem
The sides of a triangle are of lengths , , and . The altitudes of the triangle meet at point . if is the altitude to the side of length , the ratio is:
Solution 1
Using Law of Cosines and the fact that the ratio equals cos(a)/[cos(b)cos(c)] B 5:11
Solution 2 (coordinates)
The reason why we have shorter than is that all the ratios' left hand side () is less than the ratios' right hand side ().
We label point as the origin and point , logically, as . By Heron's Formula, the area of this triangle is Thus the height perpendicular to has a length of and by the Pythagorean Theorem, and have lengths and respectively. These lengths tell us that is at .
The slope of is and the slope of is by taking the negative reciprocal of Therefore, the equation of line can best be represented by
We next find the intersection of and . We automatically know the -value; it is just because is a straight line hitting Therefore, the -value is at Therefore, the intersection between and is at .
We also need to find the intersection between and . To do that, we know that the line of is represented as and the slope of line is We just need to find line 's y-intercept. So far, we have where is a real y-intercept. We know that is located at so we plug that into the equation and yield Therefore, the intersection between the two lines is After that, we use the distance formula: has a length of and has a length of Thus, we have that -OreoChocolate
Solution 3 (Ceva's Theorem)
A consequence of Ceva's Theorem that is sometimes attributed to van Aubel states that:
We must then find and . To find note that and are both right triangles sharing a common height, . Thus
Subtracting the two equations to eliminate the common height term ():
A similar computation using and gives us:
Returning to our original van Aubel equation yields:
Therefore
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See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 34 |
Followed by Problem 36 | |
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