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Difference between revisions of "1966 AHSME Problems/Problem 14"

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Draw the rectangle <math>ABCD</math> with <math>AB</math> = <math>5</math> and <math>AD</math> = <math>3</math>. We created our diagonal, <math>AC</math> and use the Pythagorean Theorem to find the length of <math>AC</math>, which is <math>\sqrt34</math>. Since <math>EF</math> breaks the diagonal into <math>3</math> equal parts, the lenght of <math>EF</math> is <math>\frac {\sqrt34}{3}</math>. The only other thing we need is the height of <math>BEF</math>. Realize that the height of <math>BEF</math> is also the height of right triangle <math>ABC</math> using <math>AC</math> as the base. The area of <math>ABC</math> is <math>\frac{15}{2}</math> (using the side lengths of the rectangle). The height of <math>ABC</math> to base <math>AC</math> is <math>\frac{15}{2}</math> divided by <math>\sqrt34 \cdot 2</math> (remember, we multiply by <math>2</math> because we are finding the height from the area of a triangle which is <math>\frac{bh}{2}</math>). That simplfies to <math>\frac{15}{\sqrt34}</math> which equal to <math>\frac{15 \cdot \sqrt34}{34}</math>. Now doing all the arithmetic, <math>\frac {\sqrt34}{3} \cdot \frac{15 \cdot \sqrt34}{34}</math> = <math> \frac {5 \cdot 3}{3 \cdot 2}</math> = <math>\frac {5}{2}</math>.
 
Draw the rectangle <math>ABCD</math> with <math>AB</math> = <math>5</math> and <math>AD</math> = <math>3</math>. We created our diagonal, <math>AC</math> and use the Pythagorean Theorem to find the length of <math>AC</math>, which is <math>\sqrt34</math>. Since <math>EF</math> breaks the diagonal into <math>3</math> equal parts, the lenght of <math>EF</math> is <math>\frac {\sqrt34}{3}</math>. The only other thing we need is the height of <math>BEF</math>. Realize that the height of <math>BEF</math> is also the height of right triangle <math>ABC</math> using <math>AC</math> as the base. The area of <math>ABC</math> is <math>\frac{15}{2}</math> (using the side lengths of the rectangle). The height of <math>ABC</math> to base <math>AC</math> is <math>\frac{15}{2}</math> divided by <math>\sqrt34 \cdot 2</math> (remember, we multiply by <math>2</math> because we are finding the height from the area of a triangle which is <math>\frac{bh}{2}</math>). That simplfies to <math>\frac{15}{\sqrt34}</math> which equal to <math>\frac{15 \cdot \sqrt34}{34}</math>. Now doing all the arithmetic, <math>\frac {\sqrt34}{3} \cdot \frac{15 \cdot \sqrt34}{34}</math> = <math> \frac {5 \cdot 3}{3 \cdot 2}</math> = <math>\frac {5}{2}</math>.
  
−
<math>\fbox{C}</math>
+
== Solution 2==
 +
Draw the rectangle <math>ABCD</math> with <math>AB</math> = <math>5</math> and <math>AD</math> = <math>3</math>. Next draw a height from vertex <math>B</math> to diagonal <math>AC</math> intersecting at <math>G</math> and similarly from <math>D</math> to <math>AC</math> intersecting at <math>J</math>. Let's call this height <math>h</math> and we can notice that <math>BG</math> = <math>DJ</math> = <math>h</math>. Since the points <math>E</math> and <math>F</math> are trisecting diagonal <math>AC</math> the bases and heights of triangles <math>BEA</math>, <math>BEF</math>, <math>BEC</math>, <math>DEA</math>, <math>DEF</math>, and <math>DEC</math> have the same areas. Hence the area of one of these triangles such as <math>BEF</math> is <math>\frac{5 \cdot 3}{2}</math> = <math>\frac {5}{2}</math>. So our answer is <math>\fbox{C}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 21:58, 12 December 2019

Problem

The length of rectangle $ABCD$ is 5 inches and its width is 3 inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is:

$\text{(A)} \frac{3}{2} \qquad \text{(B)} \frac {5}{3} \qquad \text{(C)} \frac{5}{2} \qquad \text{(D)} \frac{1}{3}\sqrt{34} \qquad \text{(E)} \frac{1}{3}\sqrt{68}$

Solution

Draw the rectangle $ABCD$ with $AB$ = $5$ and $AD$ = $3$. We created our diagonal, $AC$ and use the Pythagorean Theorem to find the length of $AC$, which is $\sqrt34$. Since $EF$ breaks the diagonal into $3$ equal parts, the lenght of $EF$ is $\frac {\sqrt34}{3}$. The only other thing we need is the height of $BEF$. Realize that the height of $BEF$ is also the height of right triangle $ABC$ using $AC$ as the base. The area of $ABC$ is $\frac{15}{2}$ (using the side lengths of the rectangle). The height of $ABC$ to base $AC$ is $\frac{15}{2}$ divided by $\sqrt34 \cdot 2$ (remember, we multiply by $2$ because we are finding the height from the area of a triangle which is $\frac{bh}{2}$). That simplfies to $\frac{15}{\sqrt34}$ which equal to $\frac{15 \cdot \sqrt34}{34}$. Now doing all the arithmetic, $\frac {\sqrt34}{3} \cdot \frac{15 \cdot \sqrt34}{34}$ = $\frac {5 \cdot 3}{3 \cdot 2}$ = $\frac {5}{2}$.

Solution 2

Draw the rectangle $ABCD$ with $AB$ = $5$ and $AD$ = $3$. Next draw a height from vertex $B$ to diagonal $AC$ intersecting at $G$ and similarly from $D$ to $AC$ intersecting at $J$. Let's call this height $h$ and we can notice that $BG$ = $DJ$ = $h$. Since the points $E$ and $F$ are trisecting diagonal $AC$ the bases and heights of triangles $BEA$, $BEF$, $BEC$, $DEA$, $DEF$, and $DEC$ have the same areas. Hence the area of one of these triangles such as $BEF$ is $\frac{5 \cdot 3}{2}$ = $\frac {5}{2}$. So our answer is $\fbox{C}$.

See also

1966 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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

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