Question
(2.) Find the line that is parallel to \( y=2 x+7 \) and passes through the point \( (2,4) \). 3. State the converse of this statement: If a triangle is equilateral, then it is equiangular.
Ask by Gibson Carter. in the United States
Jan 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**Problem 2:**
Find the line parallel to \( y = 2x + 7 \) that passes through \( (2, 4) \).
**Answer:**
\[
y = 2x
\]
**Problem 3:**
Converse of the statement: "If a triangle is equilateral, then it is equiangular."
**Answer:**
"If a triangle is equiangular, then it is equilateral."
Solution
### Problem 2
**Find the line that is parallel to \( y = 2x + 7 \) and passes through the point \( (2, 4) \).**
**Solution:**
1. **Identify the slope of the given line:**
The equation \( y = 2x + 7 \) is in slope-intercept form (\( y = mx + b \)), where:
- Slope (\( m \)) = 2
2. **Since parallel lines have the same slope, the slope of the required line is also 2.**
3. **Use the point-slope form to find the equation of the line:**
The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Plugging in the slope \( m = 2 \) and the point \( (2, 4) \):
\[
y - 4 = 2(x - 2)
\]
4. **Simplify the equation:**
\[
y - 4 = 2x - 4 \\
y = 2x
\]
**Answer:**
\[
\boxed{y = 2x}
\]
---
### Problem 3
**State the converse of the statement:**
\[
\text{If a triangle is equilateral, then it is equiangular.}
\]
**Solution:**
The **converse** of a conditional statement "If \( P \), then \( Q \)" is "If \( Q \), then \( P \)".
Applying this to the given statement:
- **Original Statement:** If a triangle is equilateral (\( P \)), then it is equiangular (\( Q \)).
- **Converse Statement:** If a triangle is equiangular (\( Q \)), then it is equilateral (\( P \)).
**Answer:**
\[
\boxed{\text{If a triangle is equiangular, then it is equilateral.}}
\]
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Simplify this solution The Deep Dive
To find a line parallel to \( y = 2x + 7 \) that passes through the point \( (2, 4) \), we know that parallel lines share the same slope. The slope of the given line is 2. Using the point-slope form, we can express the new line as \( y - 4 = 2(x - 2) \). Simplifying this gives us \( y = 2x + 0 \), or simply \( y = 2x \). The converse of the statement "If a triangle is equilateral, then it is equiangular" is "If a triangle is equiangular, then it is equilateral." This switch in the hypothesis and conclusion shows that if a triangle has all equal angles, then it must also have all sides equal. Now that's a lovely geometric relationship!
