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:

  1. Draw the triangle.
  2. Calculate the midpoint of each side.
  3. Draw the medians.
  4. Write an equation for 2 of the medians.
  5. Solve the system of equations.

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

(From Unit 6, Lesson 16.)

Problem 3

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

Triangle ABC graphed. A = -2 comma 0, B = 0 comma 4, C = 3 comma 2. median CD, D = -1 comma 2. median BF, F = 0 point 5 comma 1. median AE, E = 1 point 5 comma 3. 
(From Unit 6, Lesson 16.)

Problem 4

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

(From Unit 6, Lesson 15.)

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.  

(From Unit 6, Lesson 14.)

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.

Parabola with focus at negative 3 comma negative 2. Directrix y equals 2. Line given at y equals negative 3.
(From Unit 6, Lesson 13.)

Problem 7

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

Decreasing line that goes through the point 0 comma 1.
  1. \(y=0.25x\)
  2. \(y=2x - 4\)
  3. \(y-2 = \text-4(x-3)\)
  4. \(2y + 8x = 7\)
  5. \(x-4y=3\)
(From Unit 6, Lesson 12.)

Problem 8

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

(From Unit 6, Lesson 9.)

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.

A:

Parabola A is wider than parabola B.

B:

Parabola B is wider than parabola A.

C:

The parabolas have the same line of symmetry.

D:

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

E:

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

(From Unit 6, Lesson 7.)