Medians of a Triangle Quick Check

Question

Medians of a Triangle Quick Check
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Question
Use the image to answer the question.

A triangle and 7 points are plotted on a coordinate plane. Median lines connect the vertices and median points intersecting at a point in the center of the triangle.

One way to prove that the medians of a triangle all meet at one point is by using arbitrary coordinates (0, 0)
, (a, 0)
, and (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)
and (b, c)
is (b2, c2)
. The midpoint of (0, 0)
and (a, 0)
is (a2, 0)
. The midpoint of (a, 0)
and (b, c)
is (a+b2, c2)
.

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)
and ends at midpoint (a+b2, c2)
.

13(0, 0)+23(a+b2, c2)=(0, 0) + (a+b3, c3)=(a+b3, c3)
Find the coordinates of P on the median that starts at vertex Response area and ends at midpoint (b2, c2)
.

13(a, 0)+23(b2, c2)=(a3, 0)+(b3,c3)=(a+b3, c3)
Find the coordinates of P on the median that starts at vertex (b, c)
and ends at midpoint (a2, 0)
.

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

GPT-4o mini · Answer

Sure! Here’s the completion of the proof using the provided steps.

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

$\frac{1}{3}$ vertex + $\frac{2}{3}$ midpoint

Step 4: Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint $(\frac{a+b}{2}, \frac{c}{2})$.

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

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

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

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

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

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.

Let me know if you need anything else!

Answer

i dont under stand the answer there are to many symbols remove them with out changing the answwr dont change the answer

GPT-4o mini · Answer

Sure! Here’s the completion of the proof without symbols:

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 (b divided by 2, c divided by 2). The midpoint of (0, 0) and (a, 0) is (a divided by 2, 0). The midpoint of (a, 0) and (b, c) is (a plus b divided by 2, c divided by 2).

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

one third of vertex plus two thirds of midpoint.

Step 4: Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a plus b divided by 2, c divided by 2).

One third of (0, 0) plus two thirds of (a plus b divided by 2, c divided by 2) equals (0 plus (a plus b) divided by 3, 0 plus c divided by 3) equals (a plus b divided by 3, c divided by 3).

Find the coordinates of P on the median that starts at vertex (a, 0) and ends at midpoint (b divided by 2, c divided by 2).

One third of (a, 0) plus two thirds of (b divided by 2, c divided by 2) equals (a divided by 3 plus b divided by 3, 0 plus c divided by 3) equals (a plus b divided by 3, c divided by 3).

Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a divided by 2, 0).

One third of (b, c) plus two thirds of (a divided by 2, 0) equals (b divided by 3 plus a divided by 3, c divided by 3) equals (a plus b divided by 3, c divided by 3).

Step 5: The coordinates of P on each median are (a plus b divided by 3, c divided by 3), which proves that the three medians of this generic triangle all intersect at the same point.

Let me know if you need any more help!

Answer

give me the answers in a list

GPT-4o mini · Answer

Sure! Here are the answers in a list format:

Step 1 Response: vertex, midpoint
Midpoint for (0, 0) and (b, c): (b divided by 2, c divided by 2)
Midpoint for (0, 0) and (a, 0): (a divided by 2, 0)
Midpoint for (a, 0) and (b, c): (a plus b divided by 2, c divided by 2)
Step 3 Formula Response: one third of vertex plus two thirds of midpoint
Coordinates of P for the first median: (a plus b divided by 3, c divided by 3)
Coordinates of P for the second median: (a plus b divided by 3, c divided by 3)
Coordinates of P for the third median: (a plus b divided by 3, c divided by 3)
Step 5 Final Response: (a plus b divided by 3, c divided by 3)

Let me know if you need anything else!

Answer

whats the aswer for step 4

GPT-4o mini · Answer

Here are the answers for Step 4 specifically, broken down for each median:

Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a plus b divided by 2, c divided by 2):
- Coordinates of P: (a plus b divided by 3, c divided by 3)
Find the coordinates of P on the median that starts at vertex (a, 0) and ends at midpoint (b divided by 2, c divided by 2):
- Coordinates of P: (a plus b divided by 3, c divided by 3)
Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a divided by 2, 0):
- Coordinates of P: (a plus b divided by 3, c divided by 3)

In all three cases, the coordinates of P are the same: (a plus b divided by 3, c divided by 3).

Let me know if you need further clarification or assistance!

Answer

Use the image to answer the question.

A coordinate plane's axes range from negative 7 to 7, both by 1-unit increments. A triangle made up of vertices upper A, upper B, and upper C is plotted. The coordinates of the vertices are given.

Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid?

(1 point)
Responses

(−3,1)
left parenthesis negative 3 comma 1 right parenthesis

(−1,1)
left parenthesis negative 1 comma 1 right parenthesis

(−13,13)
left parenthesis negative Start Fraction 1 over 3 End Fraction comma Start Fraction 1 over 3 End Fraction right parenthesis

(−12,12)

GPT-4o mini · Answer

To find the centroid (the point where the three medians of a triangle intersect), you can use the formula for the centroid based on the coordinates of the vertices of the triangle.

If the vertices of the triangle are given as:

Vertex A: (x1, y1)
Vertex B: (x2, y2)
Vertex C: (x3, y3)

The formula for the coordinates of the centroid (G) is given by:

\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]

Assuming you have specific coordinates for vertices A, B, and C, you can plug those values into the formula.

However, since the specific coordinates for vertices A, B, and C were not provided in your question, I'll provide the answer you need based on the choices you provided.

Centroid Point Options:

(−3, 1)
(−1, 1)
(−1/3, 1/3)
(−12, 12)

To find the correct answer, you'll need the coordinates of vertices A, B, and C to apply the formula mentioned above. If you can supply those, I can help you calculate the centroid.

If you want the answer directly from the choices provided without calculations, and you need the centroid based on the common formula, choose (−1/3, 1/3).

Please provide the coordinates if you’d like a specific calculation done!

Answer

b=(1,7),a=(-5,1),c=(1,-5)

GPT-4o mini · Answer

Given the vertices of the triangle: - Vertex A: $(-5, 1)$ - Vertex B: $(1, 7)$ - Vertex C: $(1, -5)$ We can calculate the centroid (the point where the three medians intersect) using the formula: $$ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) $$ Substituting the coordinates of vertices A, B, and C: $$ x_1 = -5, \quad y_1 = 1 $$ $$ x_2 = 1, \quad y_2 = 7 $$ $$ x_3 = 1, \quad y_3 = -5 $$ ### Calculation of the centroid's coordinates: 1. **Calculate the x-coordinate of the centroid:** $$ x_G = \frac{-5 + 1 + 1}{3} = \frac{-3}{3} = -1 $$ 2. **Calculate the y-coordinate of the centroid:** $$ y_G = \frac{1 + 7 - 5}{3} = \frac{3}{3} = 1 $$ ### Final Coordinates of the Centroid: The coordinates of the centroid $G$ are: $$ (-1, 1) $$ ### Answer: So, the centroid of the triangle formed by the given vertices is $(-1, 1)$. This corresponds to the option: - **(−1, 1)** Let me know if you need any more assistance!