Specialized Right Triangles
Pythagorean Theorem
100
In right triangle ABC below, m∠ = 90°, m∠ = 30°, and = 6√3.
What is the length of AB?
What is 12?
Explanation: this is a 30-60-90 triangle, which has a ratio of 1:√3:2.
Divide 6√3 by √3 to get 6.
6 multiplied by 2 to get 12.
100
What is the value of x, in inches, in the right triangle below?
1) 15
2) 8
3) 34
4) 4
What is 4) 4?
Explanation: 3, 4, and 5 are a Pythagorean triple, which are the sides needed for a right triangle.
200
What is the length of the altitude of an equilateral triangle whose side has a length of 8?
What is 4√3?
Explanation: an altitude of an equilateral triangle is also a median, which is a line bisecting one side, forming a 30-60-90 triangle.
The hypotenuse (a full side) is 8, which divided by 2 is 4 units.
4 multiplied by √3 results in 4√3.
200
In the diagram below of ADB, m∠BDA = 90, AD = 5√2, and AB = 2√15.
What is the length of BD?
1) √10
2) √20
3) √50
4) √110
What is 1) √10?
Explanation: Use the Pythagorean Theorem to create an equation and solve. (2√15)2-(5√2)2=x2 - x is the side BD.
60-50=x2
x2=10
x=√10
300
What is the length of the altitude of an equilateral triangle that has a perimeter of 18?
What is 3√3?
Explanation: 18 is divided by 3 for each side of the triangle. The altitude of an equilateral triangle is the median of it as well. The definition of a median is a line running from the midpoint of one side to its opposite angle. This forms a 30-60-90 triangle, which has a ratio of 1:√3:2.
Using this ratio, divide 6 by 2 to get 3. Multiply 3 by √3 to get 3√3.
300
The rectangle shown below has a diagonal of 18.4 cm and a width of 7 cm.
To the nearest centimeter, what is the length, x, of the rectangle?
1) 11
2) 17
3) 20
4) 25
What is 2) 17?
Explanation: using the Pythagorean Theorem, set up and equation and solve.
(18.4)2 - (7)2 = x2
338.56 - 49 = x2
x2 = 289.56
x is equivalent to around 17 after finding the square root of 289.56.
400
Find the length of side x to the nearest tenth.
What is 0.7?
Explanation: this is a 45-45-90 triangle, which has a ratio of 1:1:√2. Using this ratio, divide 1 by √2 to get √2/2, which rounds to about 0.7.
400
Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards, as shown in the diagram below.
What is the length of the diagonal, in yards, that Tanya runs?
1) 50
2) 60
3) 70
4) 80
What is 1) 50?
Explanation: 30, 40, and 50 are a Pythagorean triple.
500
The figure below is a square. Find the length of side x to the nearest tenth.
What is 3.7?
Explanation: this is a 45-45-90 triangle, which has a ratio of 1:1:√2. Using this ratio, multiply √7 by √2 to get √14, which rounds to about 3.7.
500
The diagram below shows a pennant in the shape of an isosceles triangle. The equal sides each measure 13, the altitude is x + 7, and the base is 2x.
What is the length of the base?
1) 5
2) 10
3) 12
4) 24
What is 1) 5?
Explanation: plug in all of the answer choices and check if they form a Pythagorean Triple. The only answer choice that forms a Pythagorean triple is choice 1) 5, forming the triple of 5 - 12 - 13.