# ISEE Upper Level Quantitative : Triangles

## Example Questions

### Example Question #9 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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)

(b) is the greater quantity

(a) and (b) are equal

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

(a) is the greater quantity

(b) is the greater quantity

Explanation:

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 #1 : Right Triangles

Figure NOT drawn to scale.

In the above figure,  is a right angle.

What is the length of  ?

Explanation:

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,

### Example Question #41 : Triangles

Figure NOT drawn to scale.

In the above figure,  is a right angle.

What is the length of  ?

Explanation:

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 long legs to the short legs equal to each other,

By the Pythagorean Theorem.

The proportion statement becomes

### Example Question #12 : Right Triangles

Given:  with .

Which is the greater quantity?

(a)

(b)

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Explanation:

The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:

; it follows that  is obtuse, and has measure greater than

### Example Question #13 : Right Triangles

Figure NOT drawn to scale.

In the above figure,  is a right angle.

What is the length of  ?

Explanation:

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,

### Example Question #41 : Geometry

and  are right triangles, with right angles , respectively.  and .

Which is the greater quantity?

(a)

(b)

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(a) is greater.

Explanation:

Each right triangle is a  triangle, making each triangle isosceles by the Converse of the Isosceles Triangle Theorem.

Since  and  are the right triangles, the legs are , and the hypotenuses are .

By the  Theorem,  and .

, so  and subsequently, .

### Example Question #41 : Geometry

Figure NOT drawn to scale.

In the above figure,  is a right angle.

What is the ratio of the area of  to that of  ?

169 to 25

144 to 25

12 to 5

13 to 5

144 to 25

Explanation:

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller, similar triangles.

The similarity ratio of  to  can be found by determining the ratio of one pair of corresponding sides; we will use the short leg of each,  and .

is also the long leg of , so its length can be found using the Pythagorean Theorem:

The similarity ratio is therefore

.

The ratio of the areas is the square of this ratio:

- that is, 144 to 25.

### Example Question #41 : Plane Geometry

Refer to the above right triangle. Which of the following is equal to  ?

Explanation:

By the Pythagorean Theorem,

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

Given  with right angle

Which is the greater quantity?

(a)

(b)

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) and (b) are equal.

Explanation:

The sum of the measures of the angles of a triangle is , so:

This is a  triangle, so its legs  and  are congruent. The quantities are equal.

### Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

Give the length of one leg of an isosceles right triangle whose area is the same as the right triangle in the above diagram.

Explanation:

The area of a triangle is half the product of its height and its base; in a right triangle, the legs, being perpendicular, can serve as these quantites.

The triangle in the diagram has area

square inches.

An isosceles right triangle has two legs of the same length, which we will call . The area of that triangle, which is the same as that of the one in the diagram, is therefore

inches.