Task Sets: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line - Common Core Math - Grade 8

A

Recall / Warm-Up

1.

Which of the following describes a slope triangle for a line?

A.

A triangle formed by connecting any two points not on the line

B.

A right triangle formed by two points on the line, with horizontal and vertical legs

C.

A triangle formed by three collinear points on the line

D.

Any triangle whose hypotenuse is parallel to the line

2.

Two triangles are similar. Their corresponding sides are in the ratio 1 : 3. If the shorter triangle has a vertical leg of 4 and a horizontal leg of 6, what is the horizontal leg of the larger triangle?

A.

6

B.

12

C.

18

D.

24

3.

A line passes through the origin $(0, 0)$ and the point $(3, 6)$ . What is the slope of the line?

A.

$\frac{1}{2}$

B.

$3$

C.

$2$

D.

$9$

B

Fluency Practice

1.

A line passes through the origin and has slope $3$ . What is the $y$ -value when $x = 4$ ? Use the equation $y = mx$ .

2.

Two slope triangles are drawn on the same line. The smaller triangle has rise $= 2$ and run $= 3$ . The larger triangle has rise $= 4$ . What is the run of the larger triangle? (The triangles are similar.)

3.

A line passes through the origin and the point $(2, 5)$ . Write the equation of the line in the form $y = mx$ . Enter the value of $m$ as a fraction.

4.

A line has slope $2$ and $y$ -intercept $(0, 1)$ . Using the equation $y = mx + b$ , what is the $y$ -value when $x = 3$ ?

5.

The equation of a line is $y = -3x + 4$ . What are the slope and $y$ -intercept?

A.

slope $= 4$ , $y$ -intercept $= -3$

B.

slope $= -3$ , $y$ -intercept $= 4$

C.

slope $= -3$ , $y$ -intercept $= (4, 0)$

D.

slope $= 3$ , $y$ -intercept $= -4$

C

Varied Practice

$A line with two right-triangle slope triangles drawn on it, labeled with rise and run values$

1.

The diagram shows a line with two slope triangles. Triangle 1 has rise $= 2$ and run $= 3$ . Triangle 2 has rise $= 6$ and run $= 9$ . Compute the slope using Triangle 1. Enter the slope as a fraction.

2.

Two slope triangles are drawn on the same non-vertical line. Which statement best explains why both triangles must give the same slope?

A.

Both triangles have the same area.

B.

Both triangles are similar (same angles), so their sides are proportional, meaning rise/run is the same.

C.

Both triangles are congruent (same size and shape), so all their sides are equal.

D.

Both triangles share one vertex on the line, so they must be identical.

3.

Explain in your own words why the slope between any two points on a non-vertical line is always the same. Your explanation should mention similar triangles.

4.

To derive the equation of a line through the origin with slope $m$ : let $(x, y)$ be any point on the line. The slope triangle from $(0, 0)$ to $(x, y)$ has rise $= \,$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ and run $= x$ . So slope $= \frac{\text{rise}}{\text{run}} =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $= m$ . Solving for $y$ : $y =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

rise:

slope fraction:

equation:

5.

A line has slope $2$ and $y$ -intercept $(0, -1)$ . Which equation describes the line?

A.

$y = 2x$

B.

$y = 2x - 1$

C.

$y = 2x + 1$

D.

$y = -x + 2$

D

Word Problems

$A ramp line with two slope triangles: Triangle A with legs 3 and 4, Triangle B with legs 6 and 8$

1.

A coordinate grid shows a straight ramp. Two slope triangles are drawn on the ramp. Triangle A has a horizontal leg of 4 units and a vertical leg of 3 units. Triangle B has a horizontal leg of 8 units and a vertical leg of 6 units.

What is the slope of the ramp? Enter as a fraction.

$A coordinate plane showing a line from the origin (Gate) to point (4,6) (Fountain) with a slope triangle$

2.

A garden path goes from the gate at $(0, 0)$ to a fountain at $(4, 6)$ , rising at a constant rate.

1.

What is the slope of the path? Enter as a fraction.

2.

A second path is built with the same slope but starts 1 unit higher, at $y$ -intercept $(0, 1)$ . Write the equation of the second path in the form $y = mx + b$ . Enter the value of $b$ .

3.

A wheelchair ramp starts at $(0, -2)$ and rises with slope $3$ . The ramp surface can be described by the equation $y = 3x - 2$ .

What is the $y$ -value on the ramp when $x = 5$ ?

4.

A road rises from a flat starting point. It gains 4 feet of height for every 10 feet of horizontal distance traveled. The road starts at the origin $(0, 0)$ .

Using the equation $y = mx$ , what is the slope $m$ ? Enter as a fraction in simplest form.

E

Error Analysis

1.

Kenji is finding the slope of a line. He picks two pairs of points:

Pair 1: $(0, 0)$ and $(3, 6)$ → slope $= \frac{6}{3} = 2$

Pair 2: $(1, 2)$ and $(4, 8)$ → slope $= \frac{8}{1} = 8$

Kenji concludes: "The slope depends on which points you choose."

What error did Kenji make in his second calculation?

A.

Kenji used the wrong formula — he should have added the coordinates instead of dividing.

B.

Kenji divided the $y$ -value of the second point by the $x$ -value of the first point instead of finding the change in $y$ divided by the change in $x$ .

C.

Kenji should have used a larger slope triangle to get an accurate result.

D.

Kenji's calculation is correct — slope does change depending on which points you pick.

2.

Amara is deriving the equation for a line with slope $2$ and $y$ -intercept $(0, 3)$ .

She writes: "Let $(x, y)$ be any point on the line. The slope triangle from $(0, 3)$ to $(x, y)$ has rise $= y$ and run $= x$ . So slope $= \frac{y}{x} = 2$ , which gives $y = 2x$ ."

What mistake did Amara make?

A.

Amara used the wrong rise. The rise from $(0, 3)$ to $(x, y)$ is $y - 3$ , not $y$ .

B.

Amara used the wrong run. The run should be $x - 3$ , not $x$ .

C.

Amara should have used the slope triangle from the origin $(0, 0)$ , not from $(0, 3)$ .

D.

The formula $y = 2x$ is correct because the slope is $2$ .

F

Challenge / Extension

1.