# Lesson 17

Lines 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 $$P$$.

### Solution

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### 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.

### Solution

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(From Unit 6, Lesson 16.)

### Problem 3

Triangle $$ABC$$ and its medians are shown. Write an equation for median $$AE$$.

### Solution

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(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.

### Solution

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(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.

### Solution

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(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.

### Solution

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(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?

1. $$y=0.25x$$
2. $$y=2x - 4$$
3. $$y-2 = \text-4(x-3)$$
4. $$2y + 8x = 7$$
5. $$x-4y=3$$

### Solution

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(From Unit 6, Lesson 12.)

### Problem 8

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

### Solution

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(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.

### Solution

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(From Unit 6, Lesson 7.)