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Quizzes > High School Quizzes > Mathematics

Special Triangles Practice Quiz

Boost your geometry confidence with smart practice

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the Special Triangle Challenge, an interactive geometry quiz.

What is the sum of the interior angles of any triangle?
90°
180°
360°
270°
The sum of all angles in any triangle is always 180 degrees. This is a fundamental property in Euclidean geometry.
In a 45-45-90 triangle, what is the ratio of the hypotenuse to one of the equal legs?
2
1
√2
√3
A 45-45-90 triangle has legs of equal length, and the hypotenuse is obtained by multiplying one leg by √2. This ratio is a key characteristic of isosceles right triangles.
What are the angles in an equilateral triangle?
Two angles are 60° and one angle is 90°
Angles vary depending on the triangle
One angle is 90° and two angles are 45°
Each angle is 60°
An equilateral triangle has all three angles equal because all its sides are equal. Since the total sum of angles is 180°, each angle must measure 60°.
What is the side ratio in a 30-60-90 triangle when the shortest side is represented by x?
x : (x/√3) : 2x
x : x√3 : 2x
x : 2x : 3x
x : x : x√2
In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2 when the shortest side is x. This fixed ratio is critical for solving problems involving such triangles.
Which type of triangle contains a 90° angle?
Right triangle
Acute triangle
Equilateral triangle
Obtuse triangle
A triangle that contains a 90° angle is defined as a right triangle. This is one of the basic classifications of triangles based on their angles.
Given a 45-45-90 triangle with a leg measuring 5 units, what is the length of the hypotenuse?
10
5√3
5√2
5
In a 45-45-90 triangle, the hypotenuse is calculated by multiplying the leg's length by √2. Therefore, if one leg is 5 units, the hypotenuse is 5√2 units.
In a 30-60-90 triangle, if the hypotenuse measures 12 units, what is the length of the shorter leg?
12
6
4
6√3
For a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg. Therefore, if the hypotenuse is 12 units, the shorter leg must be 6 units.
What is the length of the longer leg in a 30-60-90 triangle if the shorter leg is 7 units?
7√2
7√3
7
14
In a 30-60-90 triangle, the longer leg is found by multiplying the shorter leg by √3. Thus, if the shorter leg is 7 units, the longer leg is 7√3 units.
If an equilateral triangle has a side length of 8 units, what is its height?
8
8√2
4
4√3
The height of an equilateral triangle is given by the formula h = (√3/2) × side. With a side length of 8 units, the height computes to 4√3 units.
Which statement is true regarding the altitudes of an equilateral triangle?
Only two altitudes are congruent
They are all different in length
They are all equal in length and intersect at a single point
They do not intersect at a common point
In an equilateral triangle, symmetry ensures that all altitudes are congruent. Additionally, these altitudes intersect at a single point, which serves as the centroid, incenter, and orthocenter.
In a right triangle, if one acute angle is 30°, what is the measure of the other acute angle?
30°
60°
45°
90°
A right triangle's acute angles must sum to 90° since one angle is already 90°. Therefore, if one acute angle is 30°, the other must be 60°.
Which formula correctly represents the relationship between the leg and the hypotenuse in a 45-45-90 triangle?
Hypotenuse = leg × √3
Leg = hypotenuse × √2
Hypotenuse = leg × 2
Hypotenuse = leg × √2
In a 45-45-90 triangle, the hypotenuse is determined by multiplying the length of a leg by √2. This relationship is a fundamental property of isosceles right triangles.
What is the standard side ratio for a 45-45-90 triangle?
1:√3:2
1:1:2
1:2:√3
1:1:√2
The side ratio of a 45-45-90 triangle is 1:1:√2, meaning both legs are of equal length and the hypotenuse is √2 times a leg. This ratio is widely used in solving geometric problems.
In an isosceles right triangle, what are the measures of the base angles?
60° each
90° each
30° each
45° each
An isosceles right triangle has two equal acute angles. Since the non-right angles must sum to 90°, each base angle is 45°.
What distinguishes a 30-60-90 triangle from an arbitrary right triangle?
It is always isosceles
It has fixed side ratios corresponding to its angles
It has a right angle that can change position
Its sides can vary with no fixed proportion
A 30-60-90 triangle is unique because the lengths of its sides always follow the fixed ratio 1 : √3 : 2. This specific relationship simplifies problem-solving when one side length is known.
A 30-60-90 triangle has an area of 9√3 square units. What is the length of its hypotenuse?
6√3
3√2
6√2
3√3
The area of a 30-60-90 triangle can be expressed as (x²√3)/2, where x is the length of the shorter leg. Solving for x and then doubling it gives the hypotenuse as 6√2 units.
An isosceles right triangle has an area of 50 square units. What is the length of one leg?
5
10
10√2
5√2
For an isosceles right triangle, the area is given by ½ × leg². Setting this equal to 50 results in a leg length of 10 units.
A 45-45-90 triangle has an area of 8 square units. What is the length of its hypotenuse?
4
8
2√2
4√2
First, the leg length is determined using the formula for area: ½ × a² = 8, which gives a = 4. Then, applying the 45-45-90 triangle ratio, the hypotenuse is 4√2.
In an equilateral triangle, if the median measures 9 units, what is the side length?
6√3
6
9√3
3√3
The median of an equilateral triangle is calculated by the formula (√3/2) × side. Rearranging this formula and substituting 9 for the median yields a side length of 6√3 units.
A right triangle with a 45° angle has an altitude to the hypotenuse measuring 7 units. What is the length of the hypotenuse?
14
7
7√2
14√2
In an isosceles right triangle, the altitude to the hypotenuse is equal to the leg length divided by √2. By determining the leg and then applying the 45-45-90 ratio, the hypotenuse is found to be 14 units.
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Study Outcomes

  1. Analyze the properties of special triangles, including 45-45-90 and 30-60-90 triangles.
  2. Apply geometric theorems to solve problems involving side lengths and angle measures.
  3. Demonstrate the ability to use the Pythagorean theorem in the context of special triangles.
  4. Interpret and validate geometric diagrams to support problem-solving strategies.
  5. Evaluate personal understanding of special triangle concepts and identify areas for improvement.

Special Triangles Cheat Sheet

  1. Understand the Pythagorean Theorem - Did you know that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides? This timeless rule is your secret weapon for tackling geometry puzzles from homework to exams. Once you master it, right-triangle mysteries will feel like child's play! Special Triangles: The Pythagorean Theorem | SparkNotes
  2. Master the 45°-45°-90° Triangle - This special isosceles right triangle has legs of equal length and a hypotenuse that's √2 times each leg. Memorizing the 1:1:√2 ratio will let you whip through calculations faster than a calculator. It's the ultimate shortcut when you spot those perfect 45° angles! Special Right Triangles - Formulas, Examples, FAQs
  3. Learn the 30°-60°-90° Triangle - In this triangle, the sides always follow the 1:√3:2 ratio, with the shortest side opposite 30° and the hypotenuse twice that length. Keep this pattern in your back pocket and you'll breeze through trig problems. Once you see a 30° or 60° angle, you'll know exactly what to do! Special Right Triangles - Formulas, Examples, FAQs
  4. Recognize Equilateral Triangles - All three sides and angles are equal, with each angle measuring 60°. Calculating the area is a snap using Area = (√3/4) × side². These perfectly balanced triangles pop up everywhere - from soccer balls to honeycombs - so they're worth memorizing! Equilateral triangle
  5. Understand Isosceles Triangles - At least two sides (and their opposite angles) are equal, making these triangles symmetrical and fun to work with. Remember, the base angles are always the same, which can help you solve for unknown angles in a flash. Spot that symmetry and you'll cut down your work time instantly! Special Triangles - Right Triangle, Equilateral Triangles, Isosceles Triangles
  6. Apply the Triangle Angle Sum Property - The interior angles of any triangle always add up to 180°, no matter its shape. This rule is your go-to for finding missing angles when two are given. It's like the geometry version of "what goes up must come down" - except with angles! Properties of Triangle: Triangles: Theorems and Formulas
  7. Use the Exterior Angle Theorem - An exterior angle equals the sum of its two remote interior angles, giving you a quick way to tackle tricky angle problems. This theorem is a lifesaver during proofs and standardized tests. Once you see the external angle, you can instantly pin down those hidden interior values! Exterior angle theorem
  8. Remember the Triangle Inequality Theorem - The sum of any two sides must be greater than the third side, or you don't get a triangle - simple as that! This check helps you decide if three lengths can actually form a triangle before you waste time on calculations. Think of it as geometry's "gateway rule." Properties of Triangle: Triangles: Theorems and Formulas
  9. Identify Pythagorean Triples - Triples like (3, 4, 5) or (5, 12, 13) instantly satisfy the Pythagorean Theorem, so you can spot right triangles without a calculator. Memorizing a few of these sets is like having cheat codes for geometry. Watch those classic 3-4-5 combinations pop up in real problems! Special Triangles: The Pythagorean Theorem | SparkNotes
  10. Practice with Real-World Applications - Take these triangle tricks into the field by calculating heights of trees or distances across riverbanks. Applying math to actual scenarios cements your skills and boosts your confidence. Plus, it's way more fun to see geometry in action than just on paper! Special Triangles in Geometry (examples, solutions, worksheets, videos, games, activities)
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