# ISEE Upper Level Quantitative : Triangles

## Example Questions

### Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

The perimeter of a regular octagon is 20% greater than that of the above right triangle. Which is the greater quantity?

(A) The length of one side of the octagon

(B) 3 yards

(B) is greater

(A) is greater

(A) and (B) are equal

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

(A) and (B) are equal

Explanation:

By the Pythagorean Theorem, the shorter leg has length

feet.

The perimeter of the right triangle is therefore

feet.

The octagon has perimeter 20% greater than this, or

feet.

A regular octagon has eight sides of equal length, so each side of this octagon has length

feet, which is equal to 3 yards. This makes the quantities equal.

### Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

The area of a square is equal to that of the above right triangle. Which is the greater quantity?

(A) The sidelength of the square

(B) 4 yards

(B) is greater

(A) is greater

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

(A) and (B) are equal

(B) is greater

Explanation:

By the Pythagorean Theorem, the shorter leg has length

feet.

The area of a triangle is equal to half the product of its base and height; for a right triangle, the legs can serve as these. The area of the above right triangle is

square feet.

The sidelength is the square root of this; , so . Therefore each sidelength of the square is just under 11 feet. 4 yards is 12 feet, so (B) is greater.

### Example Question #6 : How To Find The Length Of The Side Of A Right Triangle

Figure NOT drawn to scale.

Refer to the above triangle. Which is the greater quantity?

(a)

(b) 108

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

Explanation:

We can compare these numbers by comparing their squares.

By the Pythagorean Theorem,

Also,

, so .

### Example Question #51 : Plane Geometry

Consider a triangle, , in which , and . Which is the greater quantity?

(a) 55

(b)

(a) and (b) are equal

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

(a) is the greater quantity

(b) is the greater quantity

(b) is the greater quantity

Explanation:

Suppose .

By the Converse of the Pythagorean Theorem, a triangle is right if and only if the sum of the squares of the lengths of the smallest two sides is equal to the square of the longest side. Compare the quantities  and

Therefore, if

, so  is right, with the right angle opposite longest side . Thus,  is right and has degree measure 90.

However,  has degree measure greater than 90, so, as a consequence of the Converse of the Pythagorean Theorem and the SAS Inequality Theorem, it holds that .

### Example Question #51 : Plane Geometry

and  are right triangles, with right angles , respectively.

Which is the greater quantity?

(a) The perimeter of

(b) The perimeter of

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

It is impossible to tell from the information given.

Explanation:

No information is given about the legs of either triangle; therefore, no information about their perimeters can be deduced.

### Example Question #51 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Note: Figure NOT drawn to scale

Refer to the above triangle. Starting at point A, an insect walks clockwise along the sides of the triangle until he has walked 75% of the length of . What percent of the perimeter of the triangle has the insect walked?

Explanation:

By the Pythagorean Theorem, the distance from B to C, which we will call , is equal to

.

The perimeter of the triangle is

.

The insect traveled the entirety of the hypotenuse, which is 13 units long, and 75% of the longer leg, which adds 75% of 12, or  units. Therefore, the insect has traveled 22 out of the 30 units perimeter, or

of the perimeter.

### Example Question #53 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Refer to the above diagram, in which  is a right triangle with altitude . Which is the greater quantity?

(a) Four times the perimeter of

(b) Three times the perimeter of

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Explanation:

The altitude of a right triangle from the vertex of its right angle - which, here, is  - divides the triangle into two triangles similar to each other. The ratio of the hypotenuse of  to that of  (which are corresponding sides) is

,

making this the similarity ratio. The ratio of the perimeters of two similar triangles is the same as their similarity ratio; therefore, if  is the perimeter of  and  is the perimeter of , it follows that

Multiply both sides by 3:

Three times the perimeter of  is therefore equal to four times that of .

### Example Question #51 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Quantity A: The hypotenuse of a right triangle with sides  and .

Quantity B: The height of a triangle with an area of  and base of

The relationship of the quantities cannot be determined.

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The two quantities are equal.

Explanation:

Quantity A:  This is the special -- triangle, where the two sides have lengths of  and  and the hypotenuse is .

Quantity B: This triangle has an area of  and base of . The area of a triangle is , so that height must be

Quantity A and Quantity B are equal.

### Example Question #55 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Note: Figure NOT drawn to scale.

Refer to the above figure.

Which is the greater quantity?

(a)

(b)

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Explanation:

Since the shorter leg of the right triangle is half the hypotenuse, the triangle is a  triangle, with the  angle opposite the shorter leg. That makes .

### Example Question #52 : Geometry

Right triangle  has right angle .

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) is greater

Explanation:

The degree measures of the acute angles of a right triangle total 90, so we solve for  in the following equation:

(a)

(b)