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Medians and the centroid

The median to the hypotenuse · Geometry · 20 problems, easy to harder

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How we do it

  1. Use the median rule
  2. Find one part
  3. Find AG
  4. Check

The steps for Median AD = 12, find AG. Every solution names its steps, so you always know what comes next.

Worked examples

Median AD = 12, find AG Easy

The question

Question: A D is a median of triangle A B C and G is where the medians meet, with AD = 12: find AG.

1 Use the median rule

A median goes from a corner to the middle of the opposite side.

D is the middle of B C.

The three medians all meet at one point, G.

It is called the centroid.

G cuts each median into 2 parts and 1 part: the long part from the corner.

2 Find one part

A D is 3 equal parts: 2 for A G and 1 for G D.

12 divided by 3 is 4.

one part = 12 ÷ 3 = 4

Careful!

G is not the middle of the median.

A G is 2 parts out of 3.

Careful: not AG = 6 ✗

3 Find AG

AG is 2 parts: 8.

AG = 2 × 4 = 8

4 Check

Let's check: A G plus G D is 12, the whole median.

And A G is twice G D.

So it is right!

AG + GD = 8 + 4 = 12 = AD, 8 = 2 × 4

Let's look back.

First: use the median rule.

Then: find one part.

Then: find AG.

Last: check.

Answer
AG = 2 × 4 = 8

▶ Watch it solved

G is the centroid, AG = 10, find AD Medium

The question

Question: A D is a median of triangle A B C and G is where the medians meet, with AG = 10: find AD.

1 Use the median rule

A median goes from a corner to the middle of the opposite side.

D is the middle of B C.

The three medians all meet at one point, G.

It is called the centroid.

G cuts each median into 2 parts and 1 part: the long part from the corner.

2 Find one part

A G is 2 parts.

One part is 10 divided by 2: 5.

one part = 10 ÷ 2 = 5

Careful!

G is not the middle of the median.

A G is 2 parts out of 3.

Careful: not AG = 10 ✗

3 Find AD

AD is 3 parts: 15.

AD = 3 × 5 = 15

4 Check

Let's check: A G plus G D is 15, the whole median.

And A G is twice G D.

So it is right!

AG + GD = 10 + 5 = 15 = AD, 10 = 2 × 5

Let's look back.

First: use the median rule.

Then: find one part.

Then: find AD.

Last: check.

Answer
AD = 3 × 5 = 15

▶ Watch it solved

Median AD = 21 cm, find AG Harder

The question

Question: A D is a median of triangle A B C and G is where the medians meet, with AD = 21 cm: find AG.

1 Use the median rule

A median goes from a corner to the middle of the opposite side.

D is the middle of B C.

The three medians all meet at one point, G.

It is called the centroid.

G cuts each median into 2 parts and 1 part: the long part from the corner.

2 Find one part

A D is 3 equal parts: 2 for A G and 1 for G D.

21 divided by 3 is 7 cm.

one part = 21 ÷ 3 = 7 cm

Careful!

G is not the middle of the median.

A G is 2 parts out of 3.

Careful: not AG = 10.5 ✗

3 Find AG

AG is 2 parts: 14 cm.

AG = 2 × 7 = 14 cm

4 Check

Let's check: A G plus G D is 21, the whole median.

And A G is twice G D.

So it is right!

AG + GD = 14 + 7 = 21 = AD, 14 = 2 × 7

Let's look back.

First: use the median rule.

Then: find one part.

Then: find AG.

Last: check.

Answer
AG = 2 × 7 = 14 cm

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. median AD = 12, find AG
    Answer
    AG = 2 × 4 = 8
  2. median AD = 9, find GD
    Answer
    GD = 3
  3. median AD = 15 cm, find AG
    Answer
    AG = 2 × 5 = 10 cm
  4. G is the centroid, AG = 8, find GD
    Answer
    GD = 4
  5. G is the centroid, GD = 3, find AD
    Answer
    AD = 3 × 3 = 9
  6. right triangle, hypotenuse 10, find the median to it
    Answer
    Careful: only to the hypotenuse, in a right triangle
  7. right triangle, hypotenuse 14 cm, find the median to it
    Answer
    Careful: only to the hypotenuse, in a right triangle

Medium

  1. G is the centroid, AG = 10, find AD
    Answer
    AD = 3 × 5 = 15
  2. median AD = 18, find GD
    Answer
    GD = 6
  3. G is the centroid, GD = 5 cm, find AG
    Answer
    AG = 2 × 5 = 10 cm
  4. median AD = 7.5, find AG
    Answer
    AG = 2 × 2.5 = 5
  5. right triangle, median to the hypotenuse 6, find the hypotenuse
    Answer
    Careful: only to the hypotenuse, in a right triangle
  6. right triangle, hypotenuse 13, find the median to it
    Answer
    Careful: only to the hypotenuse, in a right triangle
  7. G is the centroid, AG = 7, find AD
    Answer
    AD = 3 × 3.5 = 10.5

Harder

  1. median AD = 21 cm, find AG
    Answer
    AG = 2 × 7 = 14 cm
  2. G is the centroid, GD = 2.5, find AD
    Answer
    AD = 3 × 2.5 = 7.5
  3. right triangle, median to the hypotenuse 4.5 cm, find the hypotenuse
    Answer
    Careful: only to the hypotenuse, in a right triangle
  4. G is the centroid, AG = 9, find GD
    Answer
    GD = 4.5
  5. right triangle, hypotenuse 25, find the median to it
    Answer
    Careful: only to the hypotenuse, in a right triangle
  6. median AD = 4.5, find GD
    Answer
    GD = 1.5