ATLAS Geometry Similarity, Right Triangles, and Trigonometry Practice

Similarity, Right Triangles, and Trigonometry skills

Understand similarity in terms of similarity transformations

Prove theorems involving similarity

Define trigonometric ratios and solve problems involving right triangles

Use trigonometric ratios and the Pythagorean theorem to solve problems

Standards basis

Arkansas Academic Standards for Mathematics (CCSS-M–aligned / Arkansas-adopted math standards) — Arkansas

How the ATLAS reports it

Arkansas’s statewide assessment system is ATLAS, which the ADE page says began serving in spring 2024. The official public page does not publish a math blueprint there, so the most defensible coverage map is the Arkansas math standards structure that ATLAS is aligned to.

Blueprint & weighting

ATLAS uses the Independent / state-specific framework (by grade domains structure).

Official sources

Arkansas DESE Assessment: Test Scores
Arkansas DESE home page
Arkansas DESE: 3–10 ATLAS Content Assessments

8 free questions · 0/0 correct

Practice playeasy

In a $30$ - $60$ - $90$ triangle the short leg is $10$ . What is the hypotenuse?

The hypotenuse is twice the short leg: $2 \times 10 = 20$ .

Practice playeasy

In the standard coordinate plane, how many points are both $5$ units from the origin and also exactly $3$ units from the line $x = 0$ ?

Points $5$ units from the origin satisfy $x^{2} + y^{2} = 25$ . Being exactly $3$ units from $x = 0$ means $|x| = 3$ , so $x = 3$ or $x = -3$ . Substituting gives $y^{2} = 16$ , so $y = 4$ or $y = -4$ for each $x$ -value. That yields $(3, 4)$ , $(3, -4)$ , $(-3, 4)$ , and $(-3, -4)$ : $4$ points.

Practice playeasy

In a right triangle, $\cos\alpha = \dfrac{24}{25}$ and $\tan\alpha = \dfrac{7}{24}$ . What is $\sin\alpha$ ?

$\sin\alpha = \tan\alpha \cdot \cos\alpha = \dfrac{7}{24} \cdot \dfrac{24}{25} = \dfrac{7}{25}$ .

Practice playeasy

A person who is $5$ feet tall casts a shadow $3$ feet long. At the same time, a water tower casts a shadow $24$ feet long. How tall is the water tower, in feet?

Set up $\dfrac{5}{3} = \dfrac{h}{24}$ . Cross-multiply: $3h = 120$ , so $h = 40$ feet.

Practice playeasy

In a $45$ - $45$ - $90$ triangle one leg is $20$ . What is the other leg?

Both legs of a $45$ - $45$ - $90$ triangle are equal, so the other leg is $20$ .

Practice playeasy

Parallel lines $m$ and $n$ are cut by a transversal. At one intersection, an angle measures $115^{\circ}$ . What is the measure of the corresponding angle at the other intersection?

When parallel lines are cut by a transversal, corresponding angles are congruent. The corresponding angle at the second intersection also measures $115^{\circ}$ .

Practice playeasy

In a right triangle, $\sin\alpha = \dfrac{5}{13}$ and $\tan\alpha = \dfrac{5}{12}$ . What is $\cos\alpha$ ?

$\cos\alpha = \dfrac{\sin\alpha}{\tan\alpha} = \dfrac{\dfrac{5}{13}}{\dfrac{5}{12}} = \dfrac{5}{13} \cdot \dfrac{12}{5} = \dfrac{12}{13}$ , or from the $5$ - $12$ - $13$ triangle: adjacent $12$ , hypotenuse $13$ .

Practice playeasy

A parking meter that is $5$ feet tall casts a shadow $2$ feet long. A radio antenna casts a shadow at the same time. If the antenna is $35$ feet tall, how long is its shadow, in feet?