ISEE Upper Level Quantitative : Triangles

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Question #71 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

The lengths of the hypotenuses of ten similar right triangles form an arithmetic sequence. The smallest triangle has legs of lengths 5 and 12 inches; the second-smallest triangle has a hypotenuse of length one and one half feet. 

Which of the following responses comes closest to the area of the largest triangle?

Possible Answers:

4 square feet

5 square feet

3 square feet

6 square feet

7 square feet

Correct answer:

4 square feet

Explanation:

The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:

 inches.

Let  be the lengths of the hypotenuses of the triangles in inches.  and , so their common difference is

The arithmetic sequence formula is 

The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting :

 inches. 

The largest triangle has hypotenuse of length 58 inches. Since the triangles are similar, corresponding sides are in proportion. If we let  and  be the lengths of the legs of the largest triangle, then

 

 

Similarly,

 

 

The area of a right triangle is half the product of its legs:

 square inches.

Divide this by 144 to convert to square feet:

Of the given responses, 4 square feet is the closest, and is the correct choice.

 

Example Question #72 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

In Square  is the midpoint of  is the midpoint of , and  is the midpoint of . Draw the segments  and .

Which is the greater quantity?

(a) The area of

(b) The area of  

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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

Correct answer:

(a) and (b) are equal

Explanation:

The figure referenced is below:
Square x

Let  be the common sidelength of Square .

Then .

The area of right triangle  is half the product of its legs, so 

 

, so 

The area of right triangle  is half the product of its legs, so 

 

 and  have the same area.

 

Example Question #73 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Right triangle

Figure NOT drawn to scale

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

(a) Four times the area of 

(b) Three times the area of 

Possible Answers:

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

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

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 areas of two similar triangles is the square of their similarity ratio, which here is

, or .

Therefore, if  is the area of  and  is the area of , it follows that

Four times the area of  is ; three times the area of  is

, so three times the area of  is the greater quantity.

Example Question #74 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Right triangle 2

Figure NOT drawn to scale.

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

(a) Twice the area of 

(b) The area of 

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

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

Correct answer:

(b) is the greater quantity

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. Also, since  measures 90 degrees and  measure 30 degrees,  measures 60 degrees, making  a 30-60-90 triangle.

Because of this, the ratio of the measures of the legs of  is 

,

Since these legs coincide with the hypotenuses of  and , this is also the similarity ratio of the latter to the former. The ratio of the areas is the square of this, or 

Therefore, the area of  is three times that of . This makes (b) the greater quantity.

Example Question #75 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Triangle 4

The above figure depicts Trapezoid . Which is the greater quantity?

(a) The area of 

(b) The area of 

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

The area of a triangle is one half times the product of its height and the length of its base. As can be seen in the diagram below, both  and  have height  and base of length :

Since both base length and height are the same between the two triangles, it follows that they have the same area.

Example Question #76 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Two sides of a triangle have length 8 inches and 6 inches. Which of the following lengths of the third side would make the triangle isosceles?

Possible Answers:

All of the other choices are correct.

Correct answer:

All of the other choices are correct.

Explanation:

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 6 inches or 8 inches would make the triangle meet this criterion. Also, since 6 inches and 8 inches are equal to  and , respectively, these also make the triangle isosceles. Therefore, the correct choice is that all four make the triangle isosceles.

Example Question #77 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 is an isosceles triangle with obtuse angle .

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

A triangle must have at least two acute angles; if  is obtuse, then  and  are the acute angles of . Since  is isosceles, the Isosceles Triangle Theorem requires two of the angles to be congruent; they must be the two acute angles  and . Also, the sides opposite these two angles are the congruent sides; these sides are  and , respectively. This makes the quantities (a) and (b) equal.

Example Question #78 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Isosceles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which expression is equivalent to  ?

Possible Answers:

The correct answer is not among the other choices.

Correct answer:

Explanation:

This is an isosceles triangle, so the left and right sides are of equal length. Draw the altitude of this triangle, as follows:

Isosceles

The altitude is a perpendicular bisector of the base; it is one leg of a right triangle with half the base, which is 15 inches, as the other leg, and one side, which is  inches, as the hypotenuse. By definition,

 (adjacent side divided by hypotenuse), so

 

Example Question #79 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Isosceles

Note: Figure NOT drawn to scale.

Which of the following is the greater quantity?

(A) The perimeter of the triangle

(B) 90

Possible Answers:

(A) and (B) are equal

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

(B) is greater

(A) is greater

Correct answer:

(A) is greater

Explanation:

The longest side of the triangle appears opposite the angle of greatest measure. The side of length 30 appears opposite an angle of measure .  Therefore, the sides opposite the  angles must have lengths greater than 30.

If we let this common length be , then

The perimeter of the triangle is therefore greater than 90.

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