7. The vertices of a variable triangle are (3, 4), (5cos0, 5sin0), and (5sin0,-5cos0), where θ ∈ R. The
locus of its orthocenter is
a) (x+y-1)²+(x-y-7)² = 100
c) (x+y-7)²+(x+y-1)² = 100
b) (x+y-7)²+(x-y-1)²=100
d) (x+y-7)²+(x-y+1)² = 100  for this any trian gle drawing

Here is a plot of the triangle for different values of \(\theta\), showing how the triangle changes as \(\theta\) varies. The red point represents the fixed vertex \( A(3, 4) \), while the other two vertices, \( B(5\cos\theta, 5\sin\theta) \) and \( C(5\sin\theta, -5\cos\theta) \), move according to \(\theta\).

Next, to find the locus of the orthocenter, we need to calculate the intersection point of the altitudes for each value of \(\theta\). Would you like to proceed with the algebraic solution to derive the equation of the orthocenter's locus?

Here are five questions to consider:
1. How is the slope of a line between two points calculated?
2. What is the relationship between the slopes of perpendicular lines?
3. How do you find the equation of a line given a point and a slope?
4. What is an orthocenter in a triangle, and how is it related to altitudes?
5. How does varying \(\theta\) affect the positions of the triangle's vertices?

**Tip:** The orthocenter is always the point where the three altitudes of a triangle intersect.

The vertices of a variable triangle are (3, 4), (5cosθ, 5sinθ), and (5sinθ,-5cosθ), where θ ∈ R. The locus of its orthocenter is a) (x + y - 1)² + (x - y - 7)² = 100 b) (x + y - 7)² + (x - y - 1)² = 100 c) (x + y - 7)² + (x + y - 1)² = 100 d) (x + y - 7)² + (x - y + 1)² = 100.

Explore how to find the locus of the orthocenter for a triangle with vertices (3, 4), (5cosθ, 5sinθ), and (5sinθ, -5cosθ). This problem involves concepts from coordinate geometry and trigonometry, using orthocenter and altitude intersection properties. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation