ISEE Upper Level Quantitative : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem
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Example Questions
Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
What is the hypotenuse of a right triangle with sides 5 and 8?
5√4
12
15
√89
√89
Because this is a right triangle, we can use the Pythagorean Theorem which says a2 + b2 = c2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
a2 + b2 = c2
52 + 82 = c2
25 + 64 = c2
89 = c2
c = √89
Example Question #1 : Right Triangles
Which is the greater quantity?
(a) The hypotenuse of a 
(b) The hypotenuse of a right triangle with legs of length 19 and 21
(a) is greater
(a) and (b) are equal
It is impossible to tell from the information given
(b) is greater
(b) is greater
The hypotenuses of the triangles measure as follows:
(a)
(b)
Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
Which is the greater quantity?
(a) The hypotenuse of a right triangle with legs 
(b) The hypotenuse of a right triangle with legs 
(a) and (b) are equal.
It is impossible to tell from the information given.
(b) is greater.
(a) is greater.
(a) is greater.
The hypotenuses of the triangles measure as follows:
(a)
(b)
Example Question #34 : Isee Upper Level (Grades 9 12) Quantitative Reasoning
A right triangle has a leg 
(a) The length of the second leg of the triangle
(b) 60 inches
(a) and (b) are equal.
It is impossible to tell from the information given.
(a) is greater.
(b) is greater.
(a) is greater.
The length of the second leg can be calculated using the Pythagorean Theorem. Set 
The second leg therefore measures 
Example Question #5 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?
Since we're dealing with right triangles, we can use the Pythagorean Theorem (
Example Question #6 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?
(A) The length of one side of the pentagon
(B) One and one-half feet
(A) is greater
(B) is greater
It is impossible to determine which is greater from the information given
(A) and (B) are equal
(B) is greater
By the Pythagorean Theorem, the hypotenuse of the right triangle is
The pentagon in question has sides of length 75% of 112, or
Since a pentagon has five sides of equal length, each side will have measure
One and a half feet are equivalent to 
Example Question #7 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.
Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.
Which is the greater quantity?
(A) The distance Cary runs
(B) One-fourth of a mile
(A) and (B) are equal
(B) is greater
(A) is greater
It is impossible to determine which is greater from the information given
(B) is greater
By the Pythagorean Theorem, the distance from B to C is
Cary runs
Since 5,280 feet make a mile, one-fourth of a mile is equal to
(B) is greater
Example Question #8 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
Give the length of the hypotenuse of the above right triangle in terms of 
If we let 
Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
In Square 
Which is the greater quantity?
(a)
(b)
(a) and (b) are equal
(b) is the greater quantity
(a) is the greater quantity
It cannot be determined which of (a) and (b) is greater
(b) is the greater quantity
The figure referenced is below:
For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.
Also, 
Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem
Figure NOT drawn to scale.
In the above figure, 
What is the length of 
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
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