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Quizzes > High School Quizzes > Mathematics

Special Right Triangles 45‑45‑90 Quiz: Worksheet Answers

Practice with 30‑60‑90 key and extra tips

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting 45-45-90 Mastery quiz for high school geometry students.

Which of the following is true about a 45-45-90 triangle?
The legs are congruent and the hypotenuse is √2 times the length of a leg
The triangle is always isosceles but not right
The legs are in a ratio of 1:2 with the hypotenuse
The hypotenuse is half the length of the legs
In a 45-45-90 triangle, the two legs are congruent and the hypotenuse is the leg length times √2. This is a fundamental property of this special right triangle.
Given a 45-45-90 triangle with each leg measuring 5 cm, what is the length of the hypotenuse?
2.5√2 cm
5√2 cm
10 cm
5√3 cm
Using the formula hypotenuse = leg × √2 for this triangle, the hypotenuse is 5√2 cm. This formula is central to working with 45-45-90 triangles.
Which ratio correctly represents the sides of a 45-45-90 triangle?
1 : 1 : 2
1 : √2 : 2
1 : 2 : √2
1 : 1 : √2
The sides of a 45-45-90 triangle follow the ratio 1 : 1 : √2, where the legs are equal and the hypotenuse is √2 times a leg. This ratio is essential when solving geometric problems involving this triangle.
What type of triangle is a 45-45-90 triangle?
Scalene triangle
Equilateral triangle
Acute triangle only
Isosceles right triangle
A 45-45-90 triangle is an isosceles right triangle because its two legs are congruent and it includes a 90° angle. This property distinguishes it from other types of triangles.
If one leg of a 45-45-90 triangle is 7 cm, what is the other leg's length?
7√2 cm
7/√2 cm
7 cm
14 cm
In a 45-45-90 triangle, the legs are congruent. Hence, if one leg is 7 cm, the other leg is also 7 cm.
Given a 45-45-90 triangle with hypotenuse 10 cm, what is the length of each leg?
10 cm
10√2 cm
5 cm
5√2 cm
Using the relationship leg = hypotenuse/√2, each leg is 10/√2, which simplifies to 5√2 cm. This ratio is a key property of 45-45-90 triangles.
Find the area of a 45-45-90 triangle with a leg length of 8 cm.
16 cm²
32 cm²
64 cm²
8√2 cm²
The area of a right triangle is ½ times the product of its legs. Since the legs are 8 cm each, the area is ½ × 8 × 8 = 32 cm².
If the hypotenuse of a 45-45-90 triangle is expressed as 9√2, what is the length of each leg?
9 cm
9√2 cm
18 cm
9/√2 cm
Given the hypotenuse is 9√2 and using the formula hypotenuse = leg × √2, dividing 9√2 by √2 gives 9 cm for each leg.
What is the perimeter of a 45-45-90 triangle with each leg measuring 6 cm?
12 + 6√2 cm
6 + 6√2 cm
18 cm
12 + √2 cm
The perimeter is the sum of both legs and the hypotenuse. With legs of 6 cm and hypotenuse 6√2, the perimeter is 6 + 6 + 6√2 = 12 + 6√2 cm.
How does the altitude to the hypotenuse in a 45-45-90 triangle relate to its legs?
It equals the leg length
It equals (leg * √2) / 2
It equals the hypotenuse
It equals half of the leg length
The altitude to the hypotenuse in a 45-45-90 triangle can be derived to be (leg * √2) / 2. This property is useful when solving problems involving heights and areas.
Which formula correctly determines the hypotenuse of a 45-45-90 triangle given one leg, l?
l²
l√2
2l
l/√2
The hypotenuse in a 45-45-90 triangle is obtained by multiplying the leg length by √2. This direct formula is pivotal for solving for unknown sides.
A 45-45-90 triangle has a perimeter of 34 cm. If each leg is of equal length, what is the length of one leg?
20 cm
17(2 + √2) cm
34/(2 - √2) cm
17(2 - √2) cm
Let each leg be x. The perimeter is 2x + x√2 = 34. Solving for x gives x = 34/(2+√2), which simplifies to 17(2 - √2) cm. This problem emphasizes manipulating radical expressions.
Determine the length of the median to the hypotenuse in a 45-45-90 triangle with leg length 4 cm.
4 cm
2√2 cm
2 cm
4√2 cm
In any right triangle, the median to the hypotenuse is half the length of the hypotenuse. For a leg of 4 cm, the hypotenuse is 4√2 cm, so the median is 2√2 cm.
In a 45-45-90 triangle, if the area is 50 cm², what is the length of each leg?
10 cm
50 cm
√50 cm
5√2 cm
The area of the triangle is given by ½ × leg². Setting ½ × leg² = 50 leads to leg² = 100, so each leg measures 10 cm.
Which trigonometric ratio can be used to relate the leg to the hypotenuse in a 45-45-90 triangle?
cot(45°)
sin(45°)
tan(45°)
sec(45°)
Since sin(45°) is equal to the ratio of the leg to the hypotenuse in a right triangle, it is the most direct trigonometric ratio for this relationship. This is fundamental in solving problems involving these angles.
The incircle of a 45-45-90 triangle touches all its sides. If each leg has length x, what is the area of the incircle in terms of x?
πx²(3 - 2√2)
πx²(3 - 2√2)/2
πx²(3 - √2)/2
πx²(2 - √2)
The incircle radius is r = x(2-√2)/2, so its area is πr² = π[x²(2-√2)²/4]. Simplifying (2-√2)² to (6-4√2) and dividing by 4 gives πx²(3-2√2)/2.
If a 45-45-90 triangle has an inscribed circle with an area of 2π cm², what is the length of each leg?
2(√2 + 1) cm
3√2 cm
2(√2 - 1) cm
2√2 cm
The area of the incircle gives r² = 2, so r = √2. Using the formula r = x(2-√2)/2 and solving for x yields x = 2(√2 + 1) cm.
A 45-45-90 triangle is inscribed in a circle such that the hypotenuse is the diameter. If the circle's radius is r, what is the area of the triangle in terms of r?
r²/2
2r²
πr²
r²
The hypotenuse, being the diameter, is 2r. The legs are each 2r/√2, which simplifies to r√2. The area, calculated as ½ × (r√2)², is r².
Consider a 45-45-90 triangle with leg length x that is scaled by a factor of k. How does the area of the triangle change?
It increases linearly by a factor of k
It increases by a factor of k²
It increases by a factor of k/2
It remains unchanged
Scaling the leg length by k changes the area by a factor of k² because area is proportional to the square of the side lengths. This is a basic principle in geometric scaling.
A 45-45-90 triangle has legs of length x and a hypotenuse of x√2. When rotated about one of its legs, what solid is formed and what is its volume in terms of x?
A cone with volume (1/2)πx³
A cone with volume (1/3)πx³
A sphere with volume (4/3)πx³
A cylinder with volume πx³
Rotating the triangle about one of its legs forms a cone, where the leg of rotation is the height and the other leg is the base radius. The volume is calculated as (1/3)π(radius)²(height) = (1/3)πx³.
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Study Outcomes

  1. Understand the properties of 45-45-90 triangles, including equal legs and the relationship between the legs and the hypotenuse.
  2. Apply the ratio relationships of special right triangles to calculate missing side lengths.
  3. Analyze geometric problems that involve 45-45-90 triangles to identify critical problem-solving steps.
  4. Solve exam-style questions by strategically using known triangle properties and formulas.
  5. Evaluate and correct errors in reasoning when working with special right triangle concepts.

45-45-90 & 30-60-90 Worksheet Answer Key Cheat Sheet

  1. Defining the 45-45-90 Triangle - This special right triangle boasts two 45° angles and one 90° angle, making it an isosceles right triangle with a dash of symmetry. Its equal legs create a perfectly balanced shape that's a favorite in geometry. Jump in and get to know its quirky angles! Learn more
    2. math.net
  2. Side Ratios (1:1:√2) - The secret sauce of a 45-45-90 triangle is its side ratio: 1:1:√2, meaning the two legs are twins, and the hypotenuse stretches √2 times longer than a leg. Play with these ratios to solve a ton of problems without breaking a sweat. Trust me, once you remember "One-One-Root-Two," you're golden! Check it out
    3. Byju's
  3. Finding the Hypotenuse - Got the leg length? Multiply it by √2 to uncover the hypotenuse length. For example, if each leg is 5 units, you instantly get a 5√2 wonder - no calculator gymnastics needed! See the example
    4. OnlineMathLearning
  4. Finding a Leg - If the hypotenuse is your starting point, divide it by √2 to reveal the equal legs. So, a 10√2 hypotenuse whips back into two neat 10-unit legs - pretty neat, right? Work it out
    5. OnlineMathLearning
  5. Area Formula - Crunch the numbers with Area = (leg²)/2 to find the space inside your triangle. Just square one leg, slash it by two, and voilà - you've got your area. Geometry never felt so satisfying! Calculate area
    6. SubjectMax
  6. Perimeter Formula - Add up both legs and the √2-twisted hypotenuse: Perimeter = 2 × leg + leg×√2. It's like building a road trip around the triangle - you'll quickly know how many units you're covering. Perimeter tips
    7. SubjectMax
  7. Trigonometric Relationships - In this triangle, sin(45°) and cos(45°) both equal √2/2, while tan(45°) stands tall at 1. These cool ratios come straight from the 1:1:√2 sides, making trig problems a breeze. Explore trigonometry
    8. math.net
  8. Mnemonic Magic - Remember "One-One-Root-Two" to instantly recall the leg-to-hypotenuse ratio. This catchy phrase sticks in your brain like your favorite song chorus. Sing it in math class to impress your friends (or just yourself)! Get the rhyme
    9. Basic Mathematics
  9. Square Diagonal Connection - Slice a square from corner to corner, and voilà - you get a 45-45-90 triangle. This insight helps you tackle square diagonal problems in a snap. Geometry puzzles, here you come! See it in squares
    10. Kate's Math Lessons
  10. Practice Makes Perfect - The more 45-45-90 problems you solve, the quicker you'll spot these triangles in the wild. Keep practicing to transform from triangle newbie to isosceles right-angle champion! Try some problems
    11. SchoolTube
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