Question

ladder a has a heigh of 12 feet and a base of 4 feet ladder b has a height of 18 and a base of 6 feet
Prove that the triangles formed by the two ladders are similar by comparing their corresponding side ratios
write the equation of the line representing each ladder if the bottom of each ladder starts at the origin 0,0
Pridict what happens to the slop if ladder c is introduced with a hight to base ratio of 15:5 verify your pridiction

Answer 1

To prove that the triangles formed by the two ladders are similar, we can compare their corresponding side ratios.

### Step 1: Compare Ratios of Corresponding Sides

1. **Ladder A:**
- Height = 12 feet
- Base = 4 feet
- Ratio = Height / Base = 12 / 4 = 3

2. **Ladder B:**
- Height = 18 feet
- Base = 6 feet
- Ratio = Height / Base = 18 / 6 = 3

Since both ladders have the same ratio of height to base (3), the triangles formed by the two ladders are similar by the **Angle-Angle (AA) criterion of similarity**: they have the same shape but different sizes.

### Step 2: Write the Equation of the Line for Each Ladder

Each ladder can be represented as a line in a Cartesian coordinate system, with the base at the origin (0, 0).

1. **Ladder A:**
- The line can be represented by the slope formula \(m = \frac{\text{rise}}{\text{run}} = \frac{12}{4} = 3\).
- The equation of the line (in slope-intercept form \(y = mx + b\), where \(b = 0\) because it starts at the origin) is:
\[
y = 3x
\]

2. **Ladder B:**
- The slope for Ladder B is \(m = \frac{18}{6} = 3\).
- The equation of the line is:
\[
y = 3x
\]

Both ladders have the same equation, confirming their similarity and that they rise at the same angle.

### Step 3: Introducing Ladder C

Ladder C has a height-to-base ratio of 15:5.

1. **Height of Ladder C:** 15 feet
- **Base of Ladder C:** 5 feet
- The ratio for Ladder C = Height / Base = 15 / 5 = 3.

Since Ladder C also has the same ratio of height to base (3), it will also be similar to both Ladder A and Ladder B.

### Summary and Prediction

- **Prediction:** When Ladder C is introduced, because its height-to-base ratio matches those of the other two ladders, the triangles formed by the three ladders (A, B, and C) will also be similar, and the slopes of their corresponding lines will not change.

- **Verification:** The slopes for all three ladders are:
- Ladder A: \(3\)
- Ladder B: \(3\)
- Ladder C: \(3\)

Thus, the slope remains consistent at \(3\) regardless of the introduction of Ladder C, confirming the prediction.

In conclusion, all three ladders produce similar triangles with identical slopes.