ISEE Upper Level Quantitative : Acute / Obtuse Isosceles Triangles

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

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Example Questions

Example Question #1 : Acute / Obtuse Isosceles Triangles

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 #71 : Geometry

 is an isosceles triangle with obtuse angle .

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

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 #2 : Acute / Obtuse Isosceles Triangles

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 #3 : Isosceles Triangles

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) is greater

(B) is greater

(A) and (B) are equal

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

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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