Question

In ABC, centroid D is on median AM. AD = x + 5 and DM = 3x – 5. Find AM

Answer 1

The length of side AM is $\boxed{12{\text{ units}}}.$

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 $\dfrac{2}{1}.$

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.

$\begin{aligned}{\text{AD}}&= 2\times {\text{DM}}\\x + 5&= 6x - 10\\5 + 10&=6x - x\\15&= 5x\\\frac{{15}}{5}&=x\\3&=x\\\end{aligned}$

The length of side AD can be calculated as follows,

$\begin{aligned}AD&= 3 + 5\\&= 8{\text{ units}}\\\end{aligned}$

The length of side DM can be calculated as follows,

$\begin{aligned}DM &= 3 \times 3 - 5\\&=9 - 5\\&=4\\\end{aligned}$

The length of AM can be calculated as follows,

$\begin{aligned}AM &= AD + DM\\&=8+ 4\\&= 12{\text{ units}}\\\end{aligned}$

Hence, the length of side AM is $\boxed{12{\text{ units}}}.$

Learn more:

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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.

Answer 2

Answer

Find out the value of AM.

To prove

As given

In ABC,

centroid D is on median AM.

AD = x + 5 and DM = 3x – 5.

By using the centroid property

The centroid divides each median in a ratio of 2:1.

Thus D divide the median AM  in a ratio of 2:1.

Therefore

(AD) = 2DM

(x + 5) = 6x – 10

5x = 15

$x = \frac{15}{5}$

x = 3

Now

AM = 3+ 5

     = 8 unit

DM = 3 × 3- 5

      = 4 unit

AM = AD + DM

      = 8 + 4

AM   = 12 unit

Therefore the AM is 12 unit.