# Lesson 17

Lines in Triangles

- Let’s investigate more special segments in triangles.

### Problem 1

The 3 lines \(x=3, y-2.5=\text-\frac{1}{5}(x-0.5),\) and \(y-2.5=x-3.5\) intersect at point \(P\). Find the coordinates of \(P\). Verify algebraically that the lines all intersect at \(P\).

### Problem 2

Triangle \(ABC\) has vertices at \((0,0), (5,5),\) and \((10,1)\). Kiran calculates the point of intersection of the medians using the following steps:

- Draw the triangle.
- Calculate the midpoint of each side.
- Draw the medians.
- Write an equation for 2 of the medians.
- Solve the system of equations.

Use Kiran’s method to calculate the point of intersection of the medians.

### Problem 3

Triangle \(ABC\) and its medians are shown. Write an equation for median \(AE\).

### Problem 4

Given \(A=(1,2)\) and \(B=(7,14)\), find the point that partitions segment \(AB\) in a \(2:1\) ratio.

### Problem 5

A quadrilateral has vertices \(A=(0,0), B=(4,6), C=(0,12),\) and \(D=(\text-4,6)\). Mai thinks the quadrilateral is a rhombus and Elena thinks the quadrilateral is a square. Do you agree with either of them? Show or explain your reasoning.

### Problem 6

The image shows a graph of the parabola with focus \((\text-3,\text-2)\) and directrix \(y=2\), and the line given by \(y=\text-3\). Find and verify the points where the parabola and the line intersect.

### Problem 7

For each equation, is the graph of the equation parallel to the line shown, perpendicular to the line shown, or neither?

- \(y=0.25x\)
- \(y=2x - 4\)
- \(y-2 = \text-4(x-3)\)
- \(2y + 8x = 7\)
- \(x-4y=3\)

### Problem 8

Write 2 equivalent equations for a line with \(x\)-intercept \((3,0)\) and \(y\)-intercept \((0, 2)\).

### Problem 9

Parabola A and parabola B both have the line \(y=\text-2\) as the directrix. Parabola A has its focus at \((3, 4)\) and parabola B has its focus at \((5,0)\). Select **all** true statements.

Parabola A is wider than parabola B.

Parabola B is wider than parabola A.

The parabolas have the same line of symmetry.

The line of symmetry of parabola A is to the right of that of parabola B.

The line of symmetry of parabola B is to the right of that of parabola A.