Difference between revisions of "1993 UNCO Math Contest II Problems/Problem 10"
(→Solution) |
|||
Line 23: | Line 23: | ||
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> | 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*} | \begin{align*} | ||
s &= \dfrac{51 + 52 + 53}{2} = \dfrac{3 \cdot 52}{2} = 3 \cdot 26 = 78 \ | s &= \dfrac{51 + 52 + 53}{2} = \dfrac{3 \cdot 52}{2} = 3 \cdot 26 = 78 \ | ||
Line 29: | Line 30: | ||
&= \boxed{1170} | &= \boxed{1170} | ||
\end{align*} | \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> | 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> |
Revision as of 15:06, 26 May 2017
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 |