Consider one triangle whose sides measure - units

units, and 2 units. Consider another triangle whose

sides measure 2 units, 2 units, and 2 units. Are these

triangles congruent, similar, or both? Explain your

answer.

1) Intro

Step-by-step explanation:

Sample Response: Since there are three pairs of congruent sides, we know the triangles are congruent by the SSS congruence theorem. The corresponding sides of the triangle are also in proportion, so they are also similar by the SSS similarity theorem.

The goal of this task is to estimate the measure of angles in triangles with integer side lengths. In some cases, like the equilateral triangle, this can be done using properties of the triangle. For right triangles, inverse trigonometric functions can be applied. In other cases, like part (c), an auxiliary construction is needed before applying trigonometry to calculate the angles. Since the task requires inverse trigonometric functions, students will use technology to make appropriate estimates. One very interesting side note to this task is that the algorithms that many calculators use in order to calculate complex functions such as trigonometric functions use properties of trigonometric functions along with some exact values for special angles like 30∘ and 45∘.

19

step-by-step explanation:

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1