Question Find $GE$ in a triangle where $I$ is the centroid and $DG=43$ .

Studdy Solution

STEP 1

Assumptions1. In triangle $CDE$ , $$ is the centroid. . The length of $DG$ is given as43 units.

STEP 2

The centroid of a triangle divides each median into segments in the ratio21, with the centroid being closer to the midpoint of the side (which is the point where the median intersects the side).
So, we can write the relationship between the segments of the median as $DI =2 \cdot IG$

STEP 3

Since is the centroid and $DG$ is a median, the length of $DG$ is the sum of the lengths of $DI$ and $IG$. We can write this asDG = DI + IG$$

STEP 4

Substitute the relationship from2 into the equation from3 to express $DG$ in terms of $IG$ $DG =2 \cdot IG + IG$

STEP 5

implify the equation from4 $DG =3 \cdot IG$

STEP 6

Given that $DG$ is43 units, we can substitute this into the equation from5 to find $IG$ $43 =3 \cdot IG$

STEP 7

olve the equation from6 for $IG$ $IG = \frac{43}{3}$

STEP 8

Since the centroid divides the median in the ratio21, the length of $GE$ (the segment of the median from the centroid to the vertex) is twice the length of $IG$ . We can write this as $GE =2 \cdot IG$

STEP 9

Substitute the value of $IG$ from7 into the equation from8 to find $GE$ $GE =2 \cdot \frac{43}{3}$

STEP 10

Calculate the length of $GE$ $GE =2 \cdot \frac{43}{3} = \frac{86}{3} \approx28.67$ So, the length of $GE$ is approximately28.67 units.

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Question Find $GE$ in a triangle where $I$ is the centroid and $DG=43$ .

Studdy Solution

STEP 1

Assumptions1. In triangle $CDE$ , $$ is the centroid. . The length of $DG$ is given as43 units.

STEP 2

The centroid of a triangle divides each median into segments in the ratio21, with the centroid being closer to the midpoint of the side (which is the point where the median intersects the side).
So, we can write the relationship between the segments of the median as $DI =2 \cdot IG$

STEP 3

Since is the centroid and $DG$ is a median, the length of $DG$ is the sum of the lengths of $DI$ and $IG$. We can write this asDG = DI + IG$$

STEP 4

Substitute the relationship from2 into the equation from3 to express $DG$ in terms of $IG$ $DG =2 \cdot IG + IG$

STEP 5

implify the equation from4 $DG =3 \cdot IG$

STEP 6

Given that $DG$ is43 units, we can substitute this into the equation from5 to find $IG$ $43 =3 \cdot IG$

STEP 7

olve the equation from6 for $IG$ $IG = \frac{43}{3}$

STEP 8

Since the centroid divides the median in the ratio21, the length of $GE$ (the segment of the median from the centroid to the vertex) is twice the length of $IG$ . We can write this as $GE =2 \cdot IG$

STEP 9

Substitute the value of $IG$ from7 into the equation from8 to find $GE$ $GE =2 \cdot \frac{43}{3}$

STEP 10

Calculate the length of $GE$ $GE =2 \cdot \frac{43}{3} = \frac{86}{3} \approx28.67$ So, the length of $GE$ is approximately28.67 units.

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Question Find GEGEGE in a triangle where III is the centroid and DG=43DG=43DG=43.

Studdy Solution

STEP 1

Assumptions1. In triangle CDECDECDE, $$ is the centroid. . The length of $DG$ is given as43 units.

STEP 2

The centroid of a triangle divides each median into segments in the ratio21, with the centroid being closer to the midpoint of the side (which is the point where the median intersects the side).So, we can write the relationship between the segments of the median asDI=2⋅IGDI =2 \cdot IGDI=2⋅IG

STEP 3

Since is the centroid and $DG$ is a median, the length of $DG$ is the sum of the lengths of $DI$ and $IG$. We can write this asDG = DI + IG$$

STEP 4

Substitute the relationship from2 into the equation from3 to express DGDGDG in terms of IGIGIGDG=2⋅IG+IGDG =2 \cdot IG + IGDG=2⋅IG+IG

STEP 5

implify the equation from4DG=3⋅IGDG =3 \cdot IGDG=3⋅IG

STEP 6

Given that DGDGDG is43 units, we can substitute this into the equation from5 to find IGIGIG43=3⋅IG43 =3 \cdot IG43=3⋅IG

STEP 7

olve the equation from6 for IGIGIGIG=433IG = \frac{43}{3}IG=343​

STEP 8

Since the centroid divides the median in the ratio21, the length of GEGEGE (the segment of the median from the centroid to the vertex) is twice the length of IGIGIG. We can write this asGE=2⋅IGGE =2 \cdot IGGE=2⋅IG

STEP 9

Substitute the value of IGIGIG from7 into the equation from8 to find GEGEGEGE=2⋅433GE =2 \cdot \frac{43}{3}GE=2⋅343​

STEP 10

Calculate the length of GEGEGEGE=2⋅433=863≈28.67GE =2 \cdot \frac{43}{3} = \frac{86}{3} \approx28.67GE=2⋅343​=386​≈28.67So, the length of GEGEGE is approximately28.67 units.

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Question Find $GE$ in a triangle where $I$ is the centroid and $DG=43$ .

Assumptions1. In triangle $CDE$ , $$ is the centroid. . The length of $DG$ is given as43 units.

The centroid of a triangle divides each median into segments in the ratio21, with the centroid being closer to the midpoint of the side (which is the point where the median intersects the side).
So, we can write the relationship between the segments of the median as $DI =2 \cdot IG$

Substitute the relationship from2 into the equation from3 to express $DG$ in terms of $IG$ $DG =2 \cdot IG + IG$

implify the equation from4 $DG =3 \cdot IG$

Given that $DG$ is43 units, we can substitute this into the equation from5 to find $IG$ $43 =3 \cdot IG$

olve the equation from6 for $IG$ $IG = \frac{43}{3}$

Since the centroid divides the median in the ratio21, the length of $GE$ (the segment of the median from the centroid to the vertex) is twice the length of $IG$ . We can write this as $GE =2 \cdot IG$

Substitute the value of $IG$ from7 into the equation from8 to find $GE$ $GE =2 \cdot \frac{43}{3}$

Calculate the length of $GE$ $GE =2 \cdot \frac{43}{3} = \frac{86}{3} \approx28.67$ So, the length of $GE$ is approximately28.67 units.