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ISEE Upper Level Math : ISEE Upper Level (grades 9-12) Mathematics Achievement

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

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram.

DE = 6, EC = 12

Find the length of $\overline{AB}$ .

Possible Answers:

$7\frac{3}{4}$

$7\frac{1}{4}$

$7\frac{1}{2}$

Correct answer:

$7\frac{1}{2}$

Explanation:

First, find .

Since $\overline{DE}$ is an altitude of $\Delta CDB$ from its right angle to its hypotenuse,

$\Delta DEC \sim \Delta BED$

$\frac{BE}{DE}= \frac{ED}{EC}$

$\frac{BE}{6}= \frac{6}{12}$

$BE= \frac{6}{12} \cdot 6 = 3$

CB =BE + EC = 3+ 12 = 15

$\Delta DEC \sim \Delta ABC$ by the Angle-Angle Postulate, so

$\frac{AB}{DE} = \frac{CB}{CE}$

$\frac{AB}{6} = \frac{15}{12}$

$AB= \frac{15}{12} \cdot 6$

$AB = 7\frac{1}{2}$

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Example Question #51 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Possible Answers:

$30\frac{3}{4}$

$6\frac{3}{4}$

$3\frac{3}{8}$

$15\frac{3}{8}$

Correct answer:

$3\frac{3}{8}$

Explanation:

By the Pythagorean Theorem,

$x^{2}+ 15^{2} = (x+12)^{2}$

$x^{2}+ 225 = x^{2}+24x+144$

225 = 24x+144

24x= 81

$x= 3\frac{3}{8}$

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Example Question #1 : Triangles

A right triangle $\Delta ABC$ with hypotenuse $\overline{AC}$ is inscribed in $\odot O$ , a circle with radius 26. If AB = 20 , evaluate the length of $\overline{BC}$ .

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

The arcs intercepted by a right angle are both semicircles, so hypotenuse $\overline{AC}$ shares its endpoints with two semicircles. This makes $\overline{AC}$ a diameter of the circle, and $AC = 2r = 2 \cdot 26 = 52$ .

By the Pythagorean Theorem,

$BC = \sqrt{(AC)^{2} - (AB)^{2}} = \sqrt{52^{2} -20^{2}} = \sqrt{2,704-400} = \sqrt{2,304} = 48$

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Example Question #1 : How To Find If Right Triangles Are Similar

$\Delta ABC \sim \Delta DEF$

$\angle B$ is a right angle; DE = 21 , DF = 42 .

Which is the greater quantity?

(a) $m \angle A$

(b) $60 ^{\circ }$

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

$\Delta ABC \sim \Delta DEF$ . Corresponding angles of similar triangles are congruent, so since $\angle B$ is a right angle, so is $\angle E$ .

The hypotenuse $\overline{DF}$ of $\Delta DEF$ is twice as long as leg $\overline{DE}$ ; by the $30 ^{\circ } - 60^{\circ } -90^{\circ }$ Theorem, $m \angle D = 60 ^{\circ }$ . Again, by similiarity,

$m \angle A = m \angle D = 60 ^{\circ }$ .

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Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Possible Answers:

$\sqrt{14}$

$\sqrt{65}$

$\sqrt{11}$

Correct answer:

$\sqrt{65}$

Explanation:

To solve this problem, we must utilize the Pythagorean Theorom, which states that:

$a^{2}+b^{2}=c^{2}$

We know that the base is , so we can substitute in for . We also know that the height is , so we can substitute in for .

$7^{2}+4^{2}=c^{2}$

Next we evaluate the exponents:

$7^{2}=49$

$4^{2}=16$

Now we add them together:

49+16=65

Then, $65=c^{2}$ .

is not a perfect square, so we simply write the square root as $\sqrt{65}$ .

$c=\sqrt{65}$

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Example Question #101 : Geometry

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Possible Answers:

$\sqrt{90}$

144

Correct answer:

$\sqrt{90}$

Explanation:

To solve this problem, we are going to use the Pythagorean Theorom, which states that $a^{2}+b^{2}=c^{2}$ .

We know that this particular right triangle has a base of , which can be substituted for , and a height of , which can be substituted for . If we rewrite the theorom using these numbers, we get:

$3^{2}+9^{2}=c^{2}$

Next, we evaluate the expoenents:

$3^{2}=9$

$9^{2}=81$

$9+81=c^{2}$

9+81=90

Then, $90=c^{2}$ .

To solve for , we must find the square root of . Since this is not a perfect square, our answer is simply $c=\sqrt{90}$ .

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Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

What is the hypotenuse of a right triangle with sides 5 and 8?

Possible Answers:

100

$\sqrt{89}$

undefined

Correct answer:

$\sqrt{89}$

Explanation:

According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is $a^{2} + b^{2}=c^{2}$ . Plugging in the sides, we get $5^{2}+8^{2}=c^{2}$ . Solving for , we find that the hypotenuse is $\sqrt{89}$ :

25+64=c^2

89=c^2

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Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

In a right triangle, two sides have length . Give the length of the hypotenuse in terms of .

Possible Answers:

$2\sqrt{3}t$

$\sqrt{2}t$

$2\sqrt{2}t$

$\sqrt{3}t$

Correct answer:

$2\sqrt{2}t$

Explanation:

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and side length.

$c^2=s^2+s^2\Rightarrow c^2=(2t)^2+(2t)^2\Rightarrow c^2=8t^2\Rightarrow c=\sqrt{8t^2}=2\sqrt{2}t$

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Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

In a right triangle, two sides have lengths 5 centimeters and 12 centimeters. Give the length of the hypotenuse.

Possible Answers:

$14\ cm$

$13.5\ cm$

$14.5\ cm$

$13\ cm$

$15\ cm$

Correct answer:

$13\ cm$

Explanation:

This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and , lengths of the other two sides.

$c^2=a^2+b^2\Rightarrow c^2=5^2+12^2\Rightarrow c^2=25+144=169$

$\Rightarrow c=\sqrt{169}\Rightarrow c=13\ cm$

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Example Question #1 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

In a rectangle, the width is 6 feet long and the length is 8 feet long. If a diagonal is drawn through the rectangle, from one corner to the other, how many feet long is that diagonal?

Possible Answers:

Correct answer:

Explanation:

Given that a rectangle has all right angles, drawing a diagonal will create a right triangle the legs are each 6 feet and 8 feet.

We know that in a 3-4-5 right triangle, when the legs are 3 feet and 4 feet, the hypotenuse will be 5 feet.

Given that the legs of this triangle are twice as long as those in the 3-4-5 triangle, it follows that the hypotense will also be twice as long.

Thus, the diagonal in through the rectangle creates a 6-8-10 triangle. 10 is therefore the length of the diagonal.

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : ISEE Upper Level (grades 9-12) Mathematics Achievement

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

Example Question #51 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #1 : Triangles

Example Question #1 : How To Find If Right Triangles Are Similar

Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #101 : Geometry

Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #1 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

All ISEE Upper Level Math Resources

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