Exercises: Special right triangles: 45-45-90 and 30-60-90 - Digital SAT Math

A

Recall / Warm-Up

1.

What is the ratio of sides in a 45-45-90 triangle (leg : leg : hypotenuse)?

A.

$1:1:2$

B.

$1:1:\\sqrt{2}$

C.

$1:\\sqrt{3}:2$

D.

$1:2:\\sqrt{3}$

2.

In a 30-60-90 triangle, which side is opposite the 30° angle?

A.

The hypotenuse

B.

The long leg

C.

The short leg

D.

All sides are equal

3.

A right isosceles triangle has both legs equal to 1. Use the Pythagorean theorem
to find the exact length of the hypotenuse.

Enter your answer using radical notation (e.g., sqrt(2)).

B

Fluency Practice

1.

A 45-45-90 triangle has legs of length 7. Find the length of the hypotenuse. Enter your answer in simplified radical form (e.g., 7*sqrt(2)).

2.

A 45-45-90 triangle has a hypotenuse of 10. Find the length of each leg. Enter your answer in simplified radical form.

3.

A 30-60-90 triangle has its short leg equal to 6. Find the length of the hypotenuse.

4.

A 30-60-90 triangle has its hypotenuse equal to 14. Find the length of the long leg. Enter your answer in simplified radical form.

5.

A 30-60-90 triangle has its long leg equal to $9\\sqrt{3}$ .
What is the length of the hypotenuse?

C

Varied Practice

D

Word Problems / Application

1.

A square floor tile has a diagonal of $8\\sqrt{2}$ inches.

What is the side length of the tile, in inches?

2.

An equilateral triangle has a side length of 8 cm. A vertical pole is placed at the midpoint of the base, rising straight up to the apex of the triangle (the altitude).

What is the height of the pole, in centimeters? Enter your answer in simplified radical form.

3.

A rooftop has two equal sides that meet at a 90° angle at the peak. Each side makes a 45° angle with the horizontal. The horizontal span from one wall to the other (the base of the attic triangle) is 20 feet.

1.

What is the length of one sloped roof side (from a wall top to the peak), in feet? Enter your answer in simplified radical form.

2.

What is the height of the attic (the vertical rise from the ceiling to the peak), in feet?

E

Error Analysis

1.

Maya solved this problem:

"A 30-60-90 triangle has a short leg of 5. Find the long leg and the hypotenuse."

Maya's work:

Long leg = $5 \\times 2 = 10$
Hypotenuse = $5 \\times \\sqrt{3} = 5\\sqrt{3}$

What mistake did Maya make?

A.

Maya should have used the 45-45-90 ratio instead of the 30-60-90 ratio.

B.

Maya swapped the roles of the long leg and hypotenuse — the hypotenuse is $2 \\times$ the short leg and the long leg is $\\sqrt{3} \\times$ the short leg.

C.

Maya multiplied incorrectly; $5 \\times 2 = 10$ and $5 \\times \\sqrt{3} = 5\\sqrt{3}$ are both arithmetic errors.

D.

Maya used the wrong triangle type — the given triangle is not a 30-60-90 triangle.

2.

Sam solved this problem:

"A 45-45-90 triangle has a hypotenuse of 6. Find the leg length."

Sam's work:

Leg = $6 \\times \\sqrt{2} = 6\\sqrt{2}$

What error did Sam make?

A.

Sam used the 30-60-90 ratio instead of the 45-45-90 ratio.

B.

Sam should have multiplied by $\\sqrt{3}$ instead of $\\sqrt{2}$ .

C.

Sam multiplied by $\\sqrt{2}$ instead of dividing — to go from hypotenuse to leg in a 45-45-90 triangle, divide by $\\sqrt{2}$ . The correct leg length is $3\\sqrt{2}$ .

D.

The hypotenuse should be divided by 2, giving a leg of 3.

F

Challenge / Extension

1.

A regular hexagon has a side length of 6. What is the distance between two opposite vertices (the longest diagonal)?

A.

$6\\sqrt{3}$

B.

$6\\sqrt{2}$

C.

$12$

D.

$9$

2.