Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1990 AHSME Problems/Problem 20"

(→‎Solution)
(→‎Solution)
 
Line 52: Line 52:
 
</asy>
 
</asy>
  
−
Let <math>\angle DAE = \alpha</math> and <math>\angle BAF = \beta</math>.
+
Let <math>\angle DAE = \alpha</math> and <math>\angle ADE = \beta</math>.
  
 
Since <math>\alpha + \beta = 90^\circ</math>, and <math>\alpha + \angle BAF = 90^\circ</math>, then <math>\beta = \angle BAF</math>. By the same logic, <math>\angle ABF = \alpha</math>.
 
Since <math>\alpha + \beta = 90^\circ</math>, and <math>\alpha + \angle BAF = 90^\circ</math>, then <math>\beta = \angle BAF</math>. By the same logic, <math>\angle ABF = \alpha</math>.

Latest revision as of 16:05, 23 March 2020

Problem

[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(rightanglemark(D, C, B)); MP("A",(0,0),W); MP("B",(7,4.2),N); MP("C",(10,0),E); MP("D",(3,-5),S); MP("E",(3,0),N); MP("F",(7,0),S); [/asy]

In the figure $ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $\overline{AC}$, and $\overline{DE}$ and $\overline{BF}$ are perpendicual to $\overline{AC}$. If $AE=3, DE=5,$ and $CE=7$, then $BF=$

$\text{(A) } 3.6\quad \text{(B) } 4\quad \text{(C) } 4.2\quad \text{(D) } 4.5\quad \text{(E) } 5$

Solution

Label the angles as shown in the diagram. Since $\angle DEC$ forms a linear pair with $\angle DEA$, $\angle DEA$ is a right angle.

[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.075; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, A)); draw(rightanglemark(B, F, A)); draw(rightanglemark(D, C, B)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, C)); MP("A",(0,0),W); MP("B",(7,4.2),N); MP("C",(10,0),E); MP("D",(3,-5),S); MP("E",(3,0),N); MP("F",(7,0),S); [/asy]

Let $\angle DAE = \alpha$ and $\angle ADE = \beta$.

Since $\alpha + \beta = 90^\circ$, and $\alpha + \angle BAF = 90^\circ$, then $\beta = \angle BAF$. By the same logic, $\angle ABF = \alpha$.


As a result, $\triangle AED \sim \triangle BFA$. By the same logic, $\triangle CFB \sim \triangle DEC$.

Then, $\frac{BF}{AF} = \frac{3}{5}$, and $\frac{CF}{BF} = \frac{5}{7}$.

Then, $7CF = 5BF$, and $5BF = 3AF$.

By the transitive property, $7CF = 3AF$. $AC = AF + CF = 10$, and plugging in, we get $CF = 3$.

Finally, plugging in to $\frac{CF}{BF} = \frac{5}{7}$, we get $BF = 4.2$ $\fbox{C}$

See also

1990 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1990_AHSME_Problems/Problem_20&oldid=120009"