Math / Geometry

QuestionQuestion Show Ex
In the diagram below of triangle $B C D, E$ is the midpoint of $\overline{B D}$ and $F$ is the midpoint of $\overline{C D}$ . If $E F=-21+3 x$ , and $B C=-24+4 x$ , what is the measure of $\overline{B C}$ ?

Studdy Solution

STEP 1

1. $E$ is the midpoint of $\overline{BD}$ .
2. $F$ is the midpoint of $\overline{CD}$ .
3. $EF$ is a line segment connecting midpoints $E$ and $F$ .
4. $EF = -21 + 3x$ .
5. $BC = -24 + 4x$ .
6. We need to find the measure of $\overline{BC}$ .

STEP 2

1. Use the Midsegment Theorem to relate $EF$ and $BC$ .
2. Set up an equation using the given expressions for $EF$ and $BC$ .
3. Solve for $x$ .
4. Substitute $x$ back into the expression for $BC$ to find its measure.

STEP 3

According to the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, $EF = \frac{1}{2} BC$ .

STEP 4

Using the Midsegment Theorem, set up the equation:
$EF = \frac{1}{2} BC$
Substitute the given expressions:
$-21 + 3x = \frac{1}{2}(-24 + 4x)$

STEP 5

Solve the equation for $x$ :
First, simplify the right side:
$-21 + 3x = \frac{1}{2}(-24 + 4x)$ $-21 + 3x = -12 + 2x$
Now, solve for $x$ :
$3x - 2x = -12 + 21$ $x = 9$

STEP 6

Substitute $x = 9$ back into the expression for $BC$ :
$BC = -24 + 4x$ $BC = -24 + 4(9)$ $BC = -24 + 36$ $BC = 12$
The measure of $\overline{BC}$ is:
$\boxed{12}$

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QuestionQuestion Show Ex
In the diagram below of triangle $B C D, E$ is the midpoint of $\overline{B D}$ and $F$ is the midpoint of $\overline{C D}$ . If $E F=-21+3 x$ , and $B C=-24+4 x$ , what is the measure of $\overline{B C}$ ?

Studdy Solution

STEP 1

1. $E$ is the midpoint of $\overline{BD}$ .
2. $F$ is the midpoint of $\overline{CD}$ .
3. $EF$ is a line segment connecting midpoints $E$ and $F$ .
4. $EF = -21 + 3x$ .
5. $BC = -24 + 4x$ .
6. We need to find the measure of $\overline{BC}$ .

STEP 2

1. Use the Midsegment Theorem to relate $EF$ and $BC$ .
2. Set up an equation using the given expressions for $EF$ and $BC$ .
3. Solve for $x$ .
4. Substitute $x$ back into the expression for $BC$ to find its measure.

STEP 3

According to the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, $EF = \frac{1}{2} BC$ .

STEP 4

Using the Midsegment Theorem, set up the equation:
$EF = \frac{1}{2} BC$
Substitute the given expressions:
$-21 + 3x = \frac{1}{2}(-24 + 4x)$

STEP 5

Solve the equation for $x$ :
First, simplify the right side:
$-21 + 3x = \frac{1}{2}(-24 + 4x)$ $-21 + 3x = -12 + 2x$
Now, solve for $x$ :
$3x - 2x = -12 + 21$ $x = 9$

STEP 6

Substitute $x = 9$ back into the expression for $BC$ :
$BC = -24 + 4x$ $BC = -24 + 4(9)$ $BC = -24 + 36$ $BC = 12$
The measure of $\overline{BC}$ is:
$\boxed{12}$

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QuestionQuestion Show ExIn the diagram below of triangle BCD,EB C D, EBCD,E is the midpoint of BD‾\overline{B D}BD and FFF is the midpoint of CD‾\overline{C D}CD. If EF=−21+3xE F=-21+3 xEF=−21+3x, and BC=−24+4xB C=-24+4 xBC=−24+4x, what is the measure of BC‾\overline{B C}BC ?

Studdy Solution

STEP 1

STEP 2

1. Use the Midsegment Theorem to relate EF EF EF and BC BC BC.2. Set up an equation using the given expressions for EF EF EF and BC BC BC.3. Solve for x x x.4. Substitute x x x back into the expression for BC BC BC to find its measure.

STEP 3

According to the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, EF=12BC EF = \frac{1}{2} BC EF=21​BC.

STEP 4

Using the Midsegment Theorem, set up the equation:EF=12BC EF = \frac{1}{2} BC EF=21​BCSubstitute the given expressions:−21+3x=12(−24+4x) -21 + 3x = \frac{1}{2}(-24 + 4x) −21+3x=21​(−24+4x)

STEP 5

Solve the equation for x x x:First, simplify the right side:−21+3x=12(−24+4x) -21 + 3x = \frac{1}{2}(-24 + 4x) −21+3x=21​(−24+4x) −21+3x=−12+2x -21 + 3x = -12 + 2x −21+3x=−12+2xNow, solve for x x x:3x−2x=−12+21 3x - 2x = -12 + 21 3x−2x=−12+21 x=9 x = 9 x=9

STEP 6

Substitute x=9 x = 9 x=9 back into the expression for BC BC BC:BC=−24+4x BC = -24 + 4x BC=−24+4x BC=−24+4(9) BC = -24 + 4(9) BC=−24+4(9) BC=−24+36 BC = -24 + 36 BC=−24+36 BC=12 BC = 12 BC=12 The measure of BC‾\overline{BC}BC is:12 \boxed{12} 12​

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QuestionQuestion Show Ex
In the diagram below of triangle $B C D, E$ is the midpoint of $\overline{B D}$ and $F$ is the midpoint of $\overline{C D}$ . If $E F=-21+3 x$ , and $B C=-24+4 x$ , what is the measure of $\overline{B C}$ ?

1. Use the Midsegment Theorem to relate $EF$ and $BC$ .
2. Set up an equation using the given expressions for $EF$ and $BC$ .
3. Solve for $x$ .
4. Substitute $x$ back into the expression for $BC$ to find its measure.

According to the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, $EF = \frac{1}{2} BC$ .

Using the Midsegment Theorem, set up the equation:
$EF = \frac{1}{2} BC$
Substitute the given expressions:
$-21 + 3x = \frac{1}{2}(-24 + 4x)$

Solve the equation for $x$ :
First, simplify the right side:
$-21 + 3x = \frac{1}{2}(-24 + 4x)$ $-21 + 3x = -12 + 2x$
Now, solve for $x$ :
$3x - 2x = -12 + 21$ $x = 9$

Substitute $x = 9$ back into the expression for $BC$ :
$BC = -24 + 4x$ $BC = -24 + 4(9)$ $BC = -24 + 36$ $BC = 12$
The measure of $\overline{BC}$ is:
$\boxed{12}$