One way to prove that the medians of a triangle all meet at one point is by using arbitrary coordinates (0, 0)

Question

(
0
,

0
)
, (a, 0)
(
𝑎
,

0
)
, and (b, c)
(
𝑏
,

𝑐
)
to represent the three vertices of the triangle. Drag and drop the missing pieces of each step to complete the proof that all three medians intersect at point P.

(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a Response area and ends at the Response area of the opposite side.

Step 2: Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2 and taking the sum of the y-coordinates and dividing by 2.

The midpoint between (0, 0)
(
0
,

0
)
and (b, c)
(
𝑏
,

𝑐
)
is (b2, c2)
(
𝑏
2
,

𝑐
2
)
. The midpoint of (0, 0)
(
0
,

0
)
and (a, 0)
(
𝑎
,

0
)
is (a2, 0)
(
𝑎
2
,

0
)
. The midpoint of (a, 0)
(
𝑎
,

0
)
and (b, c)
(
𝑏
,

𝑐
)
is (a+b2, c2)
(
𝑎
+
𝑏
2
,

𝑐
2
)
.

Step 3: Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula Response area.

Step 4: Find the coordinates of P on the median that starts at vertex (0, 0)
(
0
,

0
)
and ends at midpoint (a+b2, c2)
(
𝑎
+
𝑏
2
,

𝑐
2
)
.

13(0, 0)+23(a+b2, c2)=(0, 0) + (a+b3, c3)=(a+b3, c3)
1
3
(
0
,

0
)
+
2
3
(
𝑎
+
𝑏
2
,

𝑐
2
)
=
(
0
,

0
)

+

(
𝑎
+
𝑏
3
,

𝑐
3
)
=
(
𝑎
+
𝑏
3
,

𝑐
3
)

Find the coordinates of P on the median that starts at vertex Response area and ends at midpoint (b2, c2)
(
𝑏
2
,

𝑐
2
)
.

13(a, 0)+23(b2, c2)=(a3, 0)+(b3,c3)=(a+b3, c3)
1
3
(
𝑎
,

0
)
+
2
3
(
𝑏
2
,

𝑐
2
)
=
(
𝑎
3
,

0
)
+
(
𝑏
3
,
𝑐
3
)
=
(
𝑎
+
𝑏
3
,

𝑐
3
)

Find the coordinates of P on the median that starts at vertex (b, c)
(
𝑏
,

𝑐
)
and ends at midpoint (a2, 0)
(
𝑎
2
,

0
)
.

13(b, c)+23(a2, 0)=(b3,c3)+(a3,0)=(a+b3,c3)
1
3
(
𝑏
,

𝑐
)
+
2
3
(
𝑎
2
,

0
)
=
(
𝑏
3
,
𝑐
3
)
+
(
𝑎
3
,
0
)
=
(
𝑎
+
𝑏
3
,
𝑐
3
)

Step 5: The coordinates of P on each median are Response area, which proves that the three medians of this generic triangle all intersect at the same point.

Answer 1

To complete the proof that all three medians of a triangle intersect at the same point P, we will fill in the missing pieces step by step.

Step 1:

Draw the three medians of the triangle. Draw each median so that it starts at a vertex and ends at the midpoint of the opposite side.

Step 2:

Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2 and taking the sum of the y-coordinates and dividing by 2.

The midpoint between (0, 0) and (b, c) is \((\frac{b}{2}, \frac{c}{2})\). The midpoint of (0, 0) and (a, 0) is \((\frac{a}{2}, 0)\). The midpoint of (a, 0) and (b, c) is \((\frac{a+b}{2}, \frac{c}{2})\).

Step 3:

Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula \( \frac{1}{3}( \text{starting point} ) + \frac{2}{3}( \text{midpoint} )\).

Step 4:

Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint \((\frac{a+b}{2}, \frac{c}{2})\).

\(\frac{1}{3}(0, 0) + \frac{2}{3}(\frac{a+b}{2}, \frac{c}{2}) = (0, 0) + (\frac{a+b}{3}, \frac{c}{3}) = (\frac{a+b}{3}, \frac{c}{3})\)

Find the coordinates of P on the median that starts at vertex \((a, 0)\) and ends at midpoint \((\frac{b}{2}, \frac{c}{2})\).

\(\frac{1}{3}(a, 0) + \frac{2}{3}(\frac{b}{2}, \frac{c}{2}) = (\frac{a}{3}, 0) + (\frac{b}{3}, \frac{c}{3}) = (\frac{a+b}{3}, \frac{c}{3})\)

Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint \((\frac{a}{2}, 0)\).

\(\frac{1}{3}(b, c) + \frac{2}{3}(\frac{a}{2}, 0) = (\frac{b}{3}, \frac{c}{3}) + (\frac{a}{3}, 0) = (\frac{a+b}{3}, \frac{c}{3})\)

Step 5:

The coordinates of P on each median are \((\frac{a+b}{3}, \frac{c}{3})\), which proves that the three medians of this generic triangle all intersect at the same point.

This proof confirms that regardless of the specific triangle, the medians always meet at a single point, referred to as the centroid of the triangle.