### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Right Triangles

What is the hypotenuse of a right triangle with sides 5 and 8?

**Possible Answers:**

15

12

√89

5√4

**Correct answer:**

√89

Because this is a right triangle, we can use the Pythagorean Theorem which says *a*^{2} + *b*^{2} = *c*^{2}, 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.

*a*^{2} + *b*^{2} = *c*^{2}

5^{2} + 8^{2} = *c*^{2}

25 + 64 = *c*^{2}

89 = *c*^{2}

*c* = √89

### Example Question #2 : Right Triangles

Which is the greater quantity?

(a) The hypotenuse of a right triangle with a leg of length 20

(b) The hypotenuse of a right triangle with legs of length 19 and 21

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

**Correct answer:**

(b) is greater

The hypotenuses of the triangles measure as follows:

(a)

(b)

, so , making (b) the greater quantity

### 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 and .

(b) The hypotenuse of a right triangle with legs and .

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

The hypotenuses of the triangles measure as follows:

(a)

(b)

, so , making (a) the greater quantity.

### Example Question #3 : Right Triangles

A right triangle has a leg feet long and a hypotenuse feet long. Which is the greater quantity?

(a) The length of the second leg of the triangle

(b) 60 inches

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

**Correct answer:**

(a) is greater.

The length of the second leg can be calculated using the Pythagorean Theorem. Set :

The second leg therefore measures inches.

### Example Question #5 : Right Triangles

What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?

**Possible Answers:**

**Correct answer:**

Since we're dealing with right triangles, we can use the Pythagorean Theorem (). In this formula, a and b are the sides, while c is the hypotenuse. The hypotenuse of a right triangle is the longest side and the side that is opposite the right angle. Now, we can plug into our formula, which looks like this: We simplify and get . At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.

### Example Question #1 : 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

**Possible Answers:**

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

**Correct answer:**

(B) is greater

By the Pythagorean Theorem, the hypotenuse of the right triangle is

inches, making its perimeter

inches.

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

inches.

One and a half feet are equivalent to inches, so (B) is the greater quantity.

### Example Question #5 : Right Triangles

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

**Possible Answers:**

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

**Correct answer:**

(B) is greater

By the Pythagorean Theorem, the distance from B to C is

feet

Cary runs

feet

Since 5,280 feet make a mile, one-fourth of a mile is equal to

feet.

(B) is greater

### Example Question #6 : Right Triangles

Give the length of the hypotenuse of the above right triangle in terms of .

**Possible Answers:**

**Correct answer:**

If we let be the length of the hypotenuse, then by the Pythagorean theorem,

### Example Question #41 : Plane Geometry

In Square . is the midpoint of , is the midpoint of , and is the midpoint of . Construct the line segments and .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

(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.

and are midpoints of their respective sides, so , making the hypotenuse of a triangle with legs of length 2 and 2. Therefore,

.

Also, , and since is the midpoint of , . , making the hypotenuse of a triangle with legs of length 1 and 4. Therefore,

, so

### Example Question #42 : Plane Geometry

Figure NOT drawn to scale.

In the above figure, is a right angle.

What is the length of ?

**Possible Answers:**

**Correct answer:**

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,