Isosceles triangles' sides are set with equations 3x+y=0 and x-3y=0, and a point (0;5) belongs to the base of the triangle. Find the triangle's area

To find the area of the isosceles triangle, we will determine the points of intersection of the two lines \(3x + y = 0\) and \(x - 3y = 0\), which will represent the vertices on the equal sides. We also know the base of the triangle lies along a line that passes through the point \((0, 5)\). Here’s the solution step-by-step:

### Step 1: Finding Points of Intersection of the Equal Sides
The equations of the lines representing the two equal sides are:
1. \(3x + y = 0\)
2. \(x - 3y = 0\)

Solve these simultaneously to find their intersection point:
From the first equation:
\[
y = -3x
\]
Substitute \(y = -3x\) into the second equation:
\[
x - 3(-3x) = 0
\]
\[
x + 9x = 0
\]
\[
10x = 0 \Rightarrow x = 0
\]
Substitute \(x = 0\) back into \(y = -3x\):
\[
y = -3 \cdot 0 = 0
\]
Thus, the point of intersection of the two lines is \((0, 0)\), which will serve as the vertex of the isosceles triangle.

### Step 2: Finding the Points on the Base
Since the point \((0, 5)\) lies on the base of the triangle, we need to find the intersection of the lines \(3x + y = 0\) and \(x - 3y = 0\) with a horizontal line that passes through \((0, 5)\). This line is defined as \(y = 5\).

#### Finding where \(3x + y = 0\) intersects \(y = 5\)
Substitute \(y = 5\) into \(3x + y = 0\):
\[
3x + 5 = 0
\]
\[
3x = -5
\]
\[
x = -\frac{5}{3}
\]
So, one endpoint of the base is \(\left(-\frac{5}{3}, 5\right)\).

#### Finding where \(x - 3y = 0\) intersects \(y = 5\)
Substitute \(y = 5\) into \(x - 3y = 0\):
\[
x - 3 \cdot 5 = 0
\]
\[
x - 15 = 0
\]
\[
x = 15
\]
So, the other endpoint of the base is \((15, 5)\).

### Step 3: Calculating the Length of the Base
Now that we have the two endpoints of the base \(\left(-\frac{5}{3}, 5\right)\) and \((15, 5)\), we can find the length of the base:
\[
\text{Base length} = \left|15 - \left(-\frac{5}{3}\right)\right| = \left|15 + \frac{5}{3}\right|
\]
Convert \(15\) to a fraction with a denominator of 3:
\[
= \left|\frac{45}{3} + \frac{5}{3}\right| = \frac{50}{3}
\]

### Step 4: Calculating the Height
The height of the triangle is the perpendicular distance from the vertex \((0, 0)\) to the line \(y = 5\) (where the base lies). Since the vertex is at \((0, 0)\) and the base line is at \(y = 5\), the height is simply \(5\).

### Step 5: Finding the Area of the Triangle
The area \(A\) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
Substitute the base and height:
\[
A = \frac{1}{2} \times \frac{50}{3} \times 5 = \frac{250}{3} \approx 83.33
\]

### Final Answer
The area of the triangle is:
\[
\frac{250}{3} \approx 83.33
\]

Would you like further explanation on any of these steps?

---

1. How can we determine the perpendicular distance from a point to a line?
2. What are other methods for finding the area of a triangle if given vertices instead?
3. How would the solution change if the point on the base were not given?
4. Why does knowing the triangle is isosceles help in solving the problem?
5. Can this triangle's area be verified with a different method?

**Tip:** When working with coordinates, sketching a quick diagram can clarify relationships between points and lines.

Isosceles triangles' sides are set with equations 3x+y=0 and x-3y=0, and a point (0;5) belongs to the base of the triangle. Find the triangle's area.

Discover how to find the area of an isosceles triangle defined by the lines 3x + y = 0 and x - 3y = 0, with a base point at (0, 5). This step-by-step guide covers crucial concepts in geometry and coordinate systems, suitable for Grades 9-11.

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