Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equations y = mx and y = mx + b.

Sample Practice Questions (3)

In the coordinate plane, a line passes through the origin

(0, 0)

and the point

(4, 10)

. A right triangle is drawn with a vertex at the origin, a base along the x-axis of length 4, and a height of 10. If a second similar triangle is drawn starting at the origin with a base of length 2, what is the height of the second triangle, and what does this represent?

AHeight is 5; it represents the slope

m = \frac{5}{2}

BHeight is 5; it represents the y-intercept

b = 5

CHeight is 2.5; it represents the slope

m = \frac{2.5}{1}

DHeight is 20; it represents the slope

m = 10

Multiple choice · diagramSee Answer + Explanation →

Based on the provided graph, what is the equation of the line?

y = - 2 x + 2

y = 2 x + 2

y = - \frac{1}{2} x + 2

y = - 2 x - 2

Multiple choice · diagramSee Answer + Explanation →

In the coordinate plane shown, line

k

passes through the origin

(0, 0)

. Two right triangles,

△ O A B

and

△ O C D

, are drawn with their hypotenuses on line

k

. Triangle

O A B

has a height of 3 units and a base of 2 units. Triangle

O C D

has a height of 6 units and a base of 4 units. Which statement best explains why the slope of line

k

is the same when calculated using either triangle?

AThe triangles are similar by AA similarity, so the ratio of the height to the base is constant:

\frac{3}{2} = \frac{6}{4}

BThe triangles are congruent, meaning all side lengths are equal.

CThe slope is the sum of the heights divided by the sum of the bases.

DThe triangles are similar, so the slope is the product of the base and the height.

Multiple choice · diagramSee Answer + Explanation →

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More 8th Grade Standards

NY-8.NS.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats into a rational number.→NY-8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers. Locate them approximately on a number line diagram and estimate the value of expressions.→NY-8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example 3² × 3⁻⁵ = 3⁻³ = 1/27.→NY-8.EE.2Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.→

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