The length of side AM is [tex]\boxed{12{\text{ units}}}.[/tex]
Further explanation:
An altitude is a line that is perpendicular to a side and passes through opposite vertex.
The point at which all the three medians of a triangle intersect each is known as centroid of the triangle.
Median divides the triangle into two equal parts.
Given:
In triangle ABC, D is the centroid on the median AM.
The length of AD is x+5 and the length of DM is 3x-5.
Calculation:
The centroid divides the median in the ratio of [tex]\dfrac{2}{1}.[/tex]
Here, the point D is a median.
Therefore, D divides the line AM in the ratio of 2:1.
The length of AD is 2 times the length of DM.
[tex]\begin{aligned}{\text{AD}}&= 2\times {\text{DM}}\\x + 5&= 6x - 10\\5 + 10&=6x - x\\15&= 5x\\\frac{{15}}{5}&=x\\3&=x\\\end{aligned}[/tex]
The length of side AD can be calculated as follows,
[tex]\begin{aligned}AD&= 3 + 5\\&= 8{\text{ units}}\\\end{aligned}[/tex]
The length of side DM can be calculated as follows,
[tex]\begin{aligned}DM &= 3 \times 3 - 5\\&=9 - 5\\&=4\\\end{aligned}[/tex]
The length of AM can be calculated as follows,
[tex]\begin{aligned}AM &= AD + DM\\&=8+ 4\\&= 12{\text{ units}}\\\end{aligned}[/tex]
Hence, the length of side AM is [tex]\boxed{12{\text{ units}}}.[/tex]
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Triangles
Keywords: perpendicular, altitudes, point, triangle, intersect, centroid, bisectors, perpendicular bisectors, angles,angle bisectors, median, intersection, right angle triangle, equilateral triangle, obtuse, acute.
The centroid divides the median AM into a ratio of 1 : 2
The length of AM is 40
The ratio of the centroid is represented as:
[tex]\mathbf{AD : DM = 1 : 2}[/tex]
Express as fraction
[tex]\mathbf{\frac{AD }{ DM }= \frac 1 2}[/tex]
Cross multiply
[tex]\mathbf{2AD = DM}[/tex]
Substitute values for AD and DM
[tex]\mathbf{2(x + 5) = 3x - 5}[/tex]
[tex]\mathbf{2x + 10 = 3x - 5}[/tex]
Collect like terms
[tex]\mathbf{3x - 2x = 10 + 5}[/tex]
[tex]\mathbf{x = 15}[/tex]
The length of AM is:
[tex]\mathbf{AM = AD + DM}[/tex]
So, we have:
[tex]\mathbf{AM = x + 5 + 3x - 5}[/tex]
[tex]\mathbf{AM = 4x}[/tex]
Substitute 10 for x
[tex]\mathbf{AM = 4(10)}[/tex]
[tex]\mathbf{AM = 40}[/tex]
Hence, the length of AM is 40
Read more about centroids at:
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