Properties of Triangles - Centroid Worksheets - Free Printable

Let's solve the problem step by step.

We are given a triangle on a coordinate grid with vertices labeled:

- Point A is at (1, 0)
- Point B is at (-6, 0)
- Point C is at (-3, 3)

The question asks us to determine which of the following points lies on the triangle, or possibly is a vertex? But looking at the multiple-choice options:

A) (-4, 0)
B) (-2, 1)
C) (-3, 1)
D) (-4, 1)

We need to determine which of these points lies on the triangle ABC.

---

Step 1: Understand the Triangle

From the graph:

- Base AB lies along the x-axis from (-6, 0) to (1, 0).
- The top vertex C is at (-3, 3), so it's directly above the midpoint of AB.
- So this is an isosceles triangle with base AB and apex C.

We now check each option to see if it lies on any side of triangle ABC.

---

Option A: (-4, 0)

This point has y = 0, so it lies on the x-axis. Is it between A and B?

- A is at (1, 0), B is at (-6, 0)
- So the segment AB goes from x = -6 to x = 1
- x = -4 is between -6 and 1 → yes
- So (-4, 0) lies on segment AB.

✔ So this point is on the triangle

But let’s check the others to make sure.

---

Option B: (-2, 1)

Check if this point lies on any side.

Let’s consider side AC and BC.

#### Side AC: from A(1, 0) to C(-3, 3)

Find the equation of line AC.

Slope $ m = \frac{3 - 0}{-3 - 1} = \frac{3}{-4} = -\frac{3}{4} $

Equation using point-slope form from A(1, 0):

$ y - 0 = -\frac{3}{4}(x - 1) $

So $ y = -\frac{3}{4}x + \frac{3}{4} $

Now plug in x = -2:

$ y = -\frac{3}{4}(-2) + \frac{3}{4} = \frac{6}{4} + \frac{3}{4} = \frac{9}{4} = 2.25 $

But the point is (-2, 1), and y = 1 ≠ 2.25 → not on AC.

#### Side BC: from B(-6, 0) to C(-3, 3)

Slope: $ \frac{3 - 0}{-3 - (-6)} = \frac{3}{3} = 1 $

Equation: $ y - 0 = 1(x + 6) $ → $ y = x + 6 $

Plug in x = -2: y = -2 + 6 = 4 → but point is (-2, 1), y = 1 ≠ 4 → not on BC

#### Side AB: y = 0, but (-2, 1) has y = 1 → not on AB

So (-2, 1) is not on the triangle

---

Option C: (-3, 1)

Check if this lies on any side.

x = -3, y = 1

We know point C is at (-3, 3), so vertical line at x = -3

Is there a side at x = -3? Only the altitude from C to AB.

But AB is horizontal at y = 0, and C is at (-3, 3), so the altitude is from (-3, 3) down to (-3, 0). That's a vertical segment from (-3, 3) to (-3, 0).

So the line segment from C to the base is vertical at x = -3, from y = 0 to y = 3.

So (-3, 1) lies on that vertical segment.

But is that part of the triangle?

Wait — the triangle has sides AB, BC, and AC. The vertical line from C to the base is not a side unless it's explicitly drawn.

But in this case, the triangle is made up of three sides: AB, BC, and AC.

So we must check whether (-3, 1) lies on one of the sides.

Let’s check if it lies on AC or BC.

We already have:

- AC: $ y = -\frac{3}{4}x + \frac{3}{4} $
- Plug in x = -3:
$ y = -\frac{3}{4}(-3) + \frac{3}{4} = \frac{9}{4} + \frac{3}{4} = \frac{12}{4} = 3 $ → only at C

So at x = -3, AC gives y = 3 → only point C

Similarly, BC: $ y = x + 6 $

At x = -3: y = -3 + 6 = 3 → again, only point C

So the only point at x = -3 on the triangle is C itself.

But (-3, 1) is not on AC or BC, nor on AB (since y ≠ 0)

So (-3, 1) is inside the triangle? Possibly, but not on the triangle unless it's on a side.

Wait — is it on a side?

No — because the sides are:

- AB: y = 0, from x = -6 to 1
- BC: from (-6,0) to (-3,3): y = x + 6
- AC: from (1,0) to (-3,3): y = -3/4 x + 3/4

Now, check if (-3,1) lies on any of these lines.

We just saw:

- On AB? y = 0 → no
- On BC? y = x + 6 → at x = -3, y = 3 → no
- On AC? y = -3/4 x + 3/4 → at x = -3: y = -3/4*(-3) + 3/4 = 9/4 + 3/4 = 12/4 = 3 → only at C

So (-3,1) is not on any side — it's inside the triangle, but not on the boundary.

So not on the triangle

---

Option D: (-4, 1)

Check if it lies on any side.

First, y = 1, not 0 → not on AB

Check if on AC or BC.

#### Check AC: $ y = -\frac{3}{4}x + \frac{3}{4} $

Plug in x = -4:

$ y = -\frac{3}{4}(-4) + \frac{3}{4} = 3 + 0.75 = 3.75 $ → not 1 → not on AC

#### Check BC: $ y = x + 6 $

At x = -4: y = -4 + 6 = 2 → not 1 → not on BC

So not on any side.

---

Conclusion:

Only Option A: (-4, 0) lies on the triangle — specifically on the base AB.

Even though other points may be inside the triangle, the question likely wants a point that lies on the triangle, meaning on its boundary.

Therefore, the correct answer is:

> ✔ A) (-4, 0)

---

Final Answer:

A) (-4, 0) ✔