Solved on Sep 03, 2023

Find the length of CD and BF given a square ABCD with straight lines AFD, BFE, and CDE, where DF=6cm and EF=10cm.
$C D=\sqrt{(6)^2+(10)^2}=\sqrt{100}=10\mathrm{~cm}$ $B F=\sqrt{(6)^2-(10)^2}=\sqrt{4}=2\mathrm{~cm}$

STEP 1

Assumptions1. $ABCD$ is a square. . $AF$ , $BFE$ and $CDE$ are straight lines.
3. $DF =6$ cm and $EF =10$ cm.
4. Since $ABCD$ is a square, all sides are equal and all angles are right angles.

STEP 2

We notice that triangle $DF$ is a right triangle, because it is part of square $ABCD$ and $DF$ and $DE$ are sides of the square. Therefore, we can use the Pythagorean theorem to find $DE$ .
The Pythagorean theorem is $a^2 + b^2 = c^2$ where $c$ is the hypotenuse and $a$ and $b$ are the other two sides of the right triangle.

STEP 3

In triangle $DF$ , $DF$ and $EF$ are the two sides and $DE$ is the hypotenuse. So, we can write the Pythagorean theorem as $DF^2 + EF^2 = DE^2$

STEP 4

Substitute the given values of $DF$ and $EF$ into the equation.
$6^2 +10^2 = DE^2$

STEP 5

Calculate the square of $DF$ and $EF$ .
$36 +100 = DE^2$

STEP 6

Add the squares of $DF$ and $EF$ .
$136 = DE^2$

STEP 7

To find $DE$ , take the square root of both sides.
$DE = \sqrt{136}$

STEP 8

Calculate the square root of136.
$DE =11.66 \text{ cm}$

STEP 9

Since $ABCD$ is a square, all sides are equal. Therefore, $CD = DE$ .
$CD =11.66 \text{ cm}$

STEP 10

Now, let's find $BF$ . Notice that triangle $BFE$ is also a right triangle. Therefore, we can use the Pythagorean theorem again.
In triangle $BFE$ , $BE$ and $EF$ are the two sides and $BF$ is the hypotenuse. So, we can write the Pythagorean theorem as $BE^2 + EF^2 = BF^2$

STEP 11

Since $ABCD$ is a square, $BE = CD =11.66$ cm. Substitute the values of $BE$ and $EF$ into the equation.
11.66^ +10^ = BF^

STEP 12

Calculate the square of $BE$ and $EF$ .
$136 +100 = BF^2$

STEP 13

Add the squares of $BE$ and $EF$ .
$236 = BF^2$

STEP 14

To find $BF$ , take the square root of both sides.
$BF = \sqrt{236}$

STEP 15

Calculate the square root of236.
$BF =15.36 \text{ cm}$ So, $CD =11.66$ cm and $BF =15.36$ cm.

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Find the length of CD and BF given a square ABCD with straight lines AFD, BFE, and CDE, where DF=6cm and EF=10cm.
$C D=\sqrt{(6)^2+(10)^2}=\sqrt{100}=10\mathrm{~cm}$ $B F=\sqrt{(6)^2-(10)^2}=\sqrt{4}=2\mathrm{~cm}$

STEP 1

Assumptions1. $ABCD$ is a square. . $AF$ , $BFE$ and $CDE$ are straight lines.
3. $DF =6$ cm and $EF =10$ cm.
4. Since $ABCD$ is a square, all sides are equal and all angles are right angles.

STEP 2

STEP 3

In triangle $DF$ , $DF$ and $EF$ are the two sides and $DE$ is the hypotenuse. So, we can write the Pythagorean theorem as $DF^2 + EF^2 = DE^2$

STEP 4

Substitute the given values of $DF$ and $EF$ into the equation.
$6^2 +10^2 = DE^2$

STEP 5

Calculate the square of $DF$ and $EF$ .
$36 +100 = DE^2$

STEP 6

Add the squares of $DF$ and $EF$ .
$136 = DE^2$

STEP 7

To find $DE$ , take the square root of both sides.
$DE = \sqrt{136}$

STEP 8

Calculate the square root of136.
$DE =11.66 \text{ cm}$

STEP 9

Since $ABCD$ is a square, all sides are equal. Therefore, $CD = DE$ .
$CD =11.66 \text{ cm}$

STEP 10

Now, let's find $BF$ . Notice that triangle $BFE$ is also a right triangle. Therefore, we can use the Pythagorean theorem again.
In triangle $BFE$ , $BE$ and $EF$ are the two sides and $BF$ is the hypotenuse. So, we can write the Pythagorean theorem as $BE^2 + EF^2 = BF^2$

STEP 11

Since $ABCD$ is a square, $BE = CD =11.66$ cm. Substitute the values of $BE$ and $EF$ into the equation.
11.66^ +10^ = BF^

STEP 12

Calculate the square of $BE$ and $EF$ .
$136 +100 = BF^2$

STEP 13

Add the squares of $BE$ and $EF$ .
$236 = BF^2$

STEP 14

To find $BF$ , take the square root of both sides.
$BF = \sqrt{236}$

STEP 15

Calculate the square root of236.
$BF =15.36 \text{ cm}$ So, $CD =11.66$ cm and $BF =15.36$ cm.

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STEP 1

Assumptions1. ABCDABCDABCD is a square. . AFAFAF, BFEBFEBFE and CDECDECDE are straight lines.3. DF=6DF =6DF=6 cm and EF=10EF =10EF=10 cm.4. Since ABCDABCDABCD is a square, all sides are equal and all angles are right angles.

STEP 2

STEP 3

In triangle DFDFDF, DFDFDF and EFEFEF are the two sides and DEDEDE is the hypotenuse. So, we can write the Pythagorean theorem asDF2+EF2=DE2DF^2 + EF^2 = DE^2DF2+EF2=DE2

STEP 4

Substitute the given values of DFDFDF and EFEFEF into the equation.62+102=DE26^2 +10^2 = DE^262+102=DE2

STEP 5

Calculate the square of DFDFDF and EFEFEF.36+100=DE236 +100 = DE^236+100=DE2

STEP 6

Add the squares of DFDFDF and EFEFEF.136=DE2136 = DE^2136=DE2

STEP 7

To find DEDEDE, take the square root of both sides.DE=136DE = \sqrt{136}DE=136​

STEP 8

Calculate the square root of136.DE=11.66 cmDE =11.66 \text{ cm}DE=11.66 cm

STEP 9

Since ABCDABCDABCD is a square, all sides are equal. Therefore, CD=DECD = DECD=DE.CD=11.66 cmCD =11.66 \text{ cm}CD=11.66 cm

STEP 10

STEP 11

Since ABCDABCDABCD is a square, BE=CD=11.66BE = CD =11.66BE=CD=11.66 cm. Substitute the values of BEBEBE and EFEFEF into the equation.11.66^ +10^ = BF^

STEP 12

Calculate the square of BEBEBE and EFEFEF.136+100=BF2136 +100 = BF^2136+100=BF2

STEP 13

Add the squares of BEBEBE and EFEFEF.236=BF2236 = BF^2236=BF2

STEP 14

To find BFBFBF, take the square root of both sides.BF=236BF = \sqrt{236}BF=236​

STEP 15

Calculate the square root of236.BF=15.36 cmBF =15.36 \text{ cm}BF=15.36 cmSo, CD=11.66CD =11.66CD=11.66 cm and BF=15.36BF =15.36BF=15.36 cm.

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Assumptions1. $ABCD$ is a square. . $AF$ , $BFE$ and $CDE$ are straight lines.
3. $DF =6$ cm and $EF =10$ cm.
4. Since $ABCD$ is a square, all sides are equal and all angles are right angles.

In triangle $DF$ , $DF$ and $EF$ are the two sides and $DE$ is the hypotenuse. So, we can write the Pythagorean theorem as $DF^2 + EF^2 = DE^2$

Substitute the given values of $DF$ and $EF$ into the equation.
$6^2 +10^2 = DE^2$

Calculate the square of $DF$ and $EF$ .
$36 +100 = DE^2$

Add the squares of $DF$ and $EF$ .
$136 = DE^2$

To find $DE$ , take the square root of both sides.
$DE = \sqrt{136}$

Calculate the square root of136.
$DE =11.66 \text{ cm}$

Since $ABCD$ is a square, all sides are equal. Therefore, $CD = DE$ .
$CD =11.66 \text{ cm}$

Since $ABCD$ is a square, $BE = CD =11.66$ cm. Substitute the values of $BE$ and $EF$ into the equation.
11.66^ +10^ = BF^

Calculate the square of $BE$ and $EF$ .
$136 +100 = BF^2$

Add the squares of $BE$ and $EF$ .
$236 = BF^2$

To find $BF$ , take the square root of both sides.
$BF = \sqrt{236}$

Calculate the square root of236.
$BF =15.36 \text{ cm}$ So, $CD =11.66$ cm and $BF =15.36$ cm.