Solved on Jan 09, 2024

Find the lengths of $PN$ and $QP$ where $P$ is the centroid of $\triangle LMN$ and $QN=9$ .

STEP 1

Assumptions
1. Point $P$ is the centroid of $\triangle L M N$ .
2. $Q N = 9$ .
3. The centroid of a triangle divides each median in the ratio $2:1$ with the longer segment being closer to the vertex.

STEP 2

Since $P$ is the centroid, it divides the median $QN$ into two segments, $QP$ and $PN$ .

STEP 3

The centroid divides the median in the ratio $2:1$ . Let's denote the length of $QP$ as $x$ . Then $PN$ will be $\frac{x}{2}$ .

STEP 4

Since $QN$ is the sum of $QP$ and $PN$ , we can write the following equation:
$QN = QP + PN$

STEP 5

Substitute $QN$ with $9$ and express $PN$ in terms of $x$ :
$9 = x + \frac{x}{2}$

STEP 6

Combine like terms by finding a common denominator:
$9 = \frac{2x}{2} + \frac{x}{2}$

STEP 7

Add the fractions:
$9 = \frac{3x}{2}$

STEP 8

Multiply both sides of the equation by $2$ to solve for $x$ :
$18 = 3x$

STEP 9

Divide both sides by $3$ to isolate $x$ :
$x = \frac{18}{3}$

STEP 10

Calculate the value of $x$ :
$x = 6$

STEP 11

Now that we have the length of $QP$ , we can find $PN$ by dividing $QP$ by $2$ :
$PN = \frac{QP}{2}$

STEP 12

Substitute $QP$ with $6$ :
$PN = \frac{6}{2}$

STEP 13

Calculate the value of $PN$ :
$PN = 3$
$PN$ is $3$ and $QP$ is $6$ .

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Find the lengths of $PN$ and $QP$ where $P$ is the centroid of $\triangle LMN$ and $QN=9$ .

STEP 1

Assumptions
1. Point $P$ is the centroid of $\triangle L M N$ .
2. $Q N = 9$ .
3. The centroid of a triangle divides each median in the ratio $2:1$ with the longer segment being closer to the vertex.

STEP 2

Since $P$ is the centroid, it divides the median $QN$ into two segments, $QP$ and $PN$ .

STEP 3

The centroid divides the median in the ratio $2:1$ . Let's denote the length of $QP$ as $x$ . Then $PN$ will be $\frac{x}{2}$ .

STEP 4

Since $QN$ is the sum of $QP$ and $PN$ , we can write the following equation:
$QN = QP + PN$

STEP 5

Substitute $QN$ with $9$ and express $PN$ in terms of $x$ :
$9 = x + \frac{x}{2}$

STEP 6

Combine like terms by finding a common denominator:
$9 = \frac{2x}{2} + \frac{x}{2}$

STEP 7

Add the fractions:
$9 = \frac{3x}{2}$

STEP 8

Multiply both sides of the equation by $2$ to solve for $x$ :
$18 = 3x$

STEP 9

Divide both sides by $3$ to isolate $x$ :
$x = \frac{18}{3}$

STEP 10

Calculate the value of $x$ :
$x = 6$

STEP 11

Now that we have the length of $QP$ , we can find $PN$ by dividing $QP$ by $2$ :
$PN = \frac{QP}{2}$

STEP 12

Substitute $QP$ with $6$ :
$PN = \frac{6}{2}$

STEP 13

Calculate the value of $PN$ :
$PN = 3$
$PN$ is $3$ and $QP$ is $6$ .

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Find the lengths of PNPNPN and QPQPQP where PPP is the centroid of △LMN\triangle LMN△LMN and QN=9QN=9QN=9.

STEP 1

Assumptions1. Point PPP is the centroid of △LMN\triangle L M N△LMN.2. QN=9Q N = 9QN=9.3. The centroid of a triangle divides each median in the ratio 2:12:12:1 with the longer segment being closer to the vertex.

STEP 2

Since PPP is the centroid, it divides the median QNQNQN into two segments, QPQPQP and PNPNPN.

STEP 3

The centroid divides the median in the ratio 2:12:12:1. Let's denote the length of QPQPQP as xxx. Then PNPNPN will be x2\frac{x}{2}2x​.

STEP 4

Since QNQNQN is the sum of QPQPQP and PNPNPN, we can write the following equation:QN=QP+PN QN = QP + PN QN=QP+PN

STEP 5

Substitute QNQNQN with 999 and express PNPNPN in terms of xxx:9=x+x2 9 = x + \frac{x}{2} 9=x+2x​

STEP 6

Combine like terms by finding a common denominator:9=2x2+x2 9 = \frac{2x}{2} + \frac{x}{2} 9=22x​+2x​

STEP 7

Add the fractions:9=3x2 9 = \frac{3x}{2} 9=23x​

STEP 8

Multiply both sides of the equation by 222 to solve for xxx:18=3x 18 = 3x 18=3x

STEP 9

Divide both sides by 333 to isolate xxx:x=183 x = \frac{18}{3} x=318​

STEP 10

Calculate the value of xxx:x=6 x = 6 x=6

STEP 11

Now that we have the length of QPQPQP, we can find PNPNPN by dividing QPQPQP by 222:PN=QP2 PN = \frac{QP}{2} PN=2QP​

STEP 12

Substitute QPQPQP with 666:PN=62 PN = \frac{6}{2} PN=26​

STEP 13

Calculate the value of PNPNPN:PN=3 PN = 3 PN=3PNPNPN is 333 and QPQPQP is 666.

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Find the lengths of $PN$ and $QP$ where $P$ is the centroid of $\triangle LMN$ and $QN=9$ .

Assumptions
1. Point $P$ is the centroid of $\triangle L M N$ .
2. $Q N = 9$ .
3. The centroid of a triangle divides each median in the ratio $2:1$ with the longer segment being closer to the vertex.

Since $P$ is the centroid, it divides the median $QN$ into two segments, $QP$ and $PN$ .

The centroid divides the median in the ratio $2:1$ . Let's denote the length of $QP$ as $x$ . Then $PN$ will be $\frac{x}{2}$ .

Since $QN$ is the sum of $QP$ and $PN$ , we can write the following equation:
$QN = QP + PN$

Substitute $QN$ with $9$ and express $PN$ in terms of $x$ :
$9 = x + \frac{x}{2}$

Combine like terms by finding a common denominator:
$9 = \frac{2x}{2} + \frac{x}{2}$

Add the fractions:
$9 = \frac{3x}{2}$

Multiply both sides of the equation by $2$ to solve for $x$ :
$18 = 3x$

Divide both sides by $3$ to isolate $x$ :
$x = \frac{18}{3}$

Calculate the value of $x$ :
$x = 6$

Now that we have the length of $QP$ , we can find $PN$ by dividing $QP$ by $2$ :
$PN = \frac{QP}{2}$

Substitute $QP$ with $6$ :
$PN = \frac{6}{2}$

Calculate the value of $PN$ :
$PN = 3$
$PN$ is $3$ and $QP$ is $6$ .