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ISEE Upper Level Quantitative : Acute / Obtuse Isosceles Triangles

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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:

$\frac{2}{3} \textrm{ ft}$

$8\textrm{ in}$

All of the other choices are correct.

$6 \textrm{ in}$

$\frac{1}{2} \textrm{ ft}$

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 $\frac{1}{2} \textrm{ ft}$ and $\frac{2}{3} \textrm{ ft}$ , respectively, these also make the triangle isosceles. Therefore, the correct choice is that all four make the triangle isosceles.

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Example Question #71 : Geometry

$\Delta ABC$ is an isosceles triangle with obtuse angle $\angle A$ .

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 $\angle A$ is obtuse, then $\angle B$ and $\angle C$ are the acute angles of $\Delta ABC$ . Since $\Delta ABC$ is isosceles, the Isosceles Triangle Theorem requires two of the angles to be congruent; they must be the two acute angles $\angle B$ and $\angle C$ . Also, the sides opposite these two angles are the congruent sides; these sides are $\overline{AB}$ and $\overline{AC}$ , respectively. This makes the quantities (a) and (b) equal.

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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:

$\frac{15}{\sin 40^{\circ }}$

$15 \cos 40^{\circ }$

$15 \sin 40^{\circ }$

$\frac{15}{\cos 40^{\circ }}$

The correct answer is not among the other choices.

Correct answer:

$\frac{15}{\cos 40^{\circ }}$

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,

$\cos 40 ^{\circ } = \frac{15}{x}$ (adjacent side divided by hypotenuse), so

$\frac{x}{\cos 40 ^{\circ }} \cdot \cos 40 ^{\circ } =\frac{x}{\cos 40 ^{\circ }} \cdot \frac{15}{x}$

$x= \frac{15}{\cos 40 ^{\circ }}$

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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 $180 ^{\circ } - (65 ^{\circ } + 65 ^{\circ } ) = 50 ^{\circ }$ . Therefore, the sides opposite the $65^{\circ }$ angles must have lengths greater than 30.

If we let this common length be , then

s > 30

s + s > 30 + 30 = 60

P = s + s + 30 > 60 + 30 = 90

The perimeter of the triangle is therefore greater than 90.

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ISEE Upper Level Quantitative : Acute / Obtuse Isosceles Triangles

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Acute / Obtuse Isosceles Triangles

Example Question #71 : Geometry

Example Question #2 : Acute / Obtuse Isosceles Triangles

Example Question #3 : Isosceles Triangles

All ISEE Upper Level Quantitative Resources

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