Two equal sides of an isosceles triangle are given by the equations 7x - y + 3 = 0 and x + y - 3 = 0

Q: Wait, why do we use absolute values in the angle formula?

Great question! The formula $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ gives us the tangent of the acute angle between two lines. Since the base of an isosceles triangle makes the same angle with both equal sides, we are equating the *magnitudes* of these angles. The absolute value ensures we are comparing the geometric inclination, regardless of whether the slope is positive or negative.

Q: Why did the first case (taking the positive sign) result in $m^2 = -1$?

That is a fantastic observation! When we equate the expressions without the negative sign, we are essentially looking for a line that bisects the angle between the two given lines. In this specific configuration, the 'positive' case leads to an impossible geometric scenario (a complex slope), which tells us that the base must align with the *other* angular possibility—the one captured by the negative sign.

Q: Is there a faster way to do this using angle bisectors?

Yes, absolutely! The base of an isosceles triangle is perpendicular to the angle bisector of the vertex angle. You could find the angle bisectors of the two given lines, find their slopes, and then find the lines perpendicular to them passing through $(1, -10)$. However, the method we used is often more direct and less prone to calculation errors involving bisector equations.

Question

Two equal sides of an isosceles triangle are given by the equations $7x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, -10)$. Determine the equation of the third side.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer. Today, we are not just solving a coordinate geometry problem; we are uncovering the hidden symmetry of an isosceles triangle.

Imagine the coordinate plane as a canvas. We are given two lines, $7x - y + 3 = 0$ and $x + y - 3 = 0$, which form the equal sides of our triangle. Our mission is to find the third side, the base, which we know must pass through the point $P(1, -10)$.

### Decoding the Slopes

Before we dive into the geometry, we must understand the 'DNA' of these lines. By rearranging the equations into the slope-intercept form $y = mx + c$, we reveal their slopes.

For the first line, $7x - y + 3 = 0$, we find $y = 7x + 3$, giving us a slope $m_1 = 7$. For the second line, $x + y - 3 = 0$, we find $y = -x + 3$, yielding a slope $m_2 = -1$. These two numbers, $7$ and $-1$, are the keys to our kingdom.

### The Isosceles Secret

The beauty of an isosceles triangle lies in its symmetry. The base makes the exact same angle with both equal sides.

If we let the slope of our unknown base be $m$, we can use the angle formula:
$$	an 	heta = \left| \frac{m_a - m_b}{1 + m_a m_b} ight|$$

By equating the angle between the base and the first side to the angle between the base and the second side, we establish the equation:
$$\left| \frac{m - 7}{1 + 7m} ight| = \left| \frac{m - (-1)}{1 + m(-1)} ight|$$

### The Algebra Battle

Because we are dealing with absolute values, we must branch into two cases.

In Case 1, we assume the expressions are equal without the negative sign. Expanding this leads us to $8m^2 + 8 = 0$, or $m^2 = -1$. As you know, the square of a real number cannot be negative, so this path leads to a dead end—a beautiful reminder that geometry often guides us away from impossible solutions.

In Case 2, we take the negative sign:
$$\frac{m - 7}{1 + 7m} = -\frac{m + 1}{1 - m}$$

Cross-multiplying and simplifying, we arrive at the quadratic equation $3m^2 + 8m - 3 = 0$. Factoring this, we find $(3m - 1)(m + 3) = 0$, giving us two possible slopes: $m = \frac{1}{3}$ and $m = -3$.

### The Final Construction

Now, we simply use the point-slope form $y - y_1 = m(x - x_1)$ with our point $P(1, -10)$.

For $m = \frac{1}{3}$, we get $y + 10 = \frac{1}{3}(x - 1)$, which simplifies to **$x - 3y - 31 = 0$**.

For $m = -3$, we get $y + 10 = -3(x - 1)$, which simplifies to **$3x + y + 7 = 0$**.

You have done it! You have successfully navigated the symmetry of the triangle to find the two possible equations for the base. Keep this analytical rigor in your toolkit; it will serve you well in every challenge to come.

Animated Solution for Mathematics - Straight Lines: Two equal sides of an isosceles triangle are given by the equations $7 x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, - 10)$ . Determine the equation of the third side.

Visualized Solution

Visualizing the Given Information

Calculating Slopes $m_{1}$ and $m_{2}$

Applying the Isosceles Property

Setting up the Angle Formula

Substituting the Slopes

Solving for $m$ : Case 1

Simplifying Case 1

Solving for $m$ : Case 2

Simplifying Case 2

Finding the Slopes

Determining the Final Equations

The Sigma Insight: Angle Between Two Lines

Analyzing the Setup

Decoding the Slopes

The Isosceles Secret

The Algebra Battle

The Final Construction

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