Difference between revisions of "1993 UNCO Math Contest II Problems/Problem 10"
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== Solution == | == Solution == | ||
+ | |||
+ | We will solve both parts at once since it is easy to get the two answers from this all-inclusive solution. | ||
+ | |||
+ | Heron's Formula states that in a triangle with sides <math>a, b, c</math> and <math>s = \frac{a + b + c}{2},</math> the area is given by <cmath>\sqrt{s(s - a)(s - b)(s - c)}.</cmath> We plug in <math>a = 52, b = 53, c = 51.</math> | ||
+ | |||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | s &= \dfrac{51 + 52 + 53}{2} = \dfrac{3 \cdot 52}{2} = 3 \cdot 26 = 78 \\ | ||
+ | [ABC] &= \sqrt{78(78 - 51)(78 - 52)(78 - 53)} = \sqrt{78(27)(26)(25)} = 5 \sqrt{(3 \cdot 26)\left(3^3\right)(2 \cdot 13)} \\ | ||
+ | &= 5 \cdot 3 \sqrt{3 \cdot 2 \cdot 13 \cdot 3 \cdot 2 \cdot 13} = 15 \sqrt{2^2 \cdot 3^2 \cdot 13^2} = 15 \sqrt{(2 \cdot 3 \cdot 13)}^2 \\ | ||
+ | &= \boxed{1170} | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Since <math>[ABC] = \frac{bh}{2},</math> we know that <math>1170 = \frac{AD \cdot 52}{2} = 26 \cdot AD.</math> Solving, we get <math>AD = 45.</math> Remembering our 8-15-17 Pythagorean triple, we see that <math>BD = \boxed{24}.</math> <math>\square</math> | ||
== See also == | == See also == | ||
{{UNCO Math Contest box|year=1993|n=II|num-b=9|after=Last Question}} | {{UNCO Math Contest box|year=1993|n=II|num-b=9|after=Last Question}} | ||
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] |
Latest revision as of 23:57, 3 June 2022
Problem
The scalene triangle has side lengths
is perpendicular to
(a) Determine the length of
(b) Determine the area of triangle
Solution
We will solve both parts at once since it is easy to get the two answers from this all-inclusive solution.
Heron's Formula states that in a triangle with sides and
the area is given by
We plug in
Since we know that
Solving, we get
Remembering our 8-15-17 Pythagorean triple, we see that
See also
1993 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |