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ISEE Upper Level Math : Triangles

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Example Questions

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

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above map. A farmer owns a triangular plot of land flanked by the three highways as shown. The farmer uses Highway 2 frequently; however, he can only access it by driving five miles north on Highway 32 or twelve miles east on Highway 100.

He wants to build a dirt road that directly accesses Highway 2. He figures that it will cost $1,500 per tenth of a mile to construct the road. By his estimate, which answer will come closest to the total cost of the shortest possible road?

Possible Answers:

$\$50,000$

$\$80,000$

$\$70,000$

$\$60,000$

$\$40,000$

Correct answer:

$\$70,000$

Explanation:

The three roads form the legs and the hypotenuse of a right triangle with legs 5 and 12 miles; by the Pythagorean Theorem, the hypotenuse is

$\sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt {169}= 13$ miles.

To find the length of the shortest possible road, which must be perpendicular to Highway 2, it should be observed that this road serves as an altitude from the vertex of the triangle to the hypotenuse.

The area of a right triangle can be calculated two ways - by taking half the product of the lengths of the legs or by taking half the product of the length of the altitude (height) and that of the hypotenuse. Setting as the height, we can solve for in the equation:

$\frac{1}{2} \cdot 13 \cdot h = \frac{1}{2} \cdot 5 \cdot 12$

6.5 h = 30

$h = 30 \div 6.5 \approx 4.6$ miles.

This is 46 tenths of a mile, so the approximate cost of the road in dollars will be

$46 \times 1,500 = 69,000$

Among the five choices, $70,000 comes closest.

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Example Question #31 : Triangles

Which of the following is true about a triangle with two angles that measure $45^{\circ}$ and $45^{\circ}$ ?

Possible Answers:

This triangle is scalene and right.

This is an equilateral triangle.

This is a right isosceles triangle.

This triangle is scalene.

This triangle is obtuse.

Correct answer:

This is a right isosceles triangle.

Explanation:

The sum of the two given angles is 90 degrees, which means that the third angle should be a right angle (90 degrees). We also know that two of the angles are equal. Therefore, the triangle is right and isosceles.

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Example Question #71 : Isee Upper Level (Grades 9 12) Mathematics Achievement

What is the value of x in a right triangle if the two acute angles are equal to 5x and 25x?

Possible Answers:

Correct answer:

Explanation:

In a right triangle, there is one right angle of 90 degrees, while the two acute angles add up to 90 degrees.

Given that the two acute angles are equal to 5x and 25x, the value of x can be calculated with the equation below:

5x+25x=90

30x=90

x=3

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

One angle of a right triangle measures 45 degrees, and the hypotenuse measures 8 centimeters. Give the area of the triangle.

Possible Answers:

$16\ cm^2$

$32\ cm^2$

$64\ cm^2$

$8\ cm^2$

$4\ cm^2$

Correct answer:

$16\ cm^2$

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 side length.

$c^2=s^2+s^2\Rightarrow c^2=2s^2\Rightarrow 8^2=2s^2\Rightarrow s^2=32$

$\Rightarrow s=\sqrt{32}=\sqrt{16\times 2}=4\sqrt{2}\ cm$

We can then plug this side length into the formula for area.

$area=\frac{s^2}{2}=\frac{(4\sqrt{2})^2}{2}=\frac{32}{2}=16\ cm^2$

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Example Question #71 : Plane Geometry

The legs of a right triangle measure $2 \frac{2}{5}$ and $3 \frac{1}{5}$ . What is its perimeter?

Possible Answers:

$9 \frac{1}{5}$

$9 \frac{3}{5}$

$8 \frac{3}{5}$

$10\frac{1}{5}$

Correct answer:

$9 \frac{3}{5}$

Explanation:

The hypotenuse of the triangle can be calculated using the Pythagorean Theorem. Set $a = 2 \frac{2}{5} = \frac{12}{5} , b=3 \frac{1}{5}= \frac{16}{5}$ :

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

$c ^{2} = \left ( \frac{12}{5} \right ) ^{2} + \left ( \frac{16}{5} \right ) ^{2}$

$c ^{2} = \frac{144}{25} + \frac{256}{25}$

$c ^{2} = \frac{400}{25} = 16$

$c = \sqrt{16} = 4$

Add the three sidelengths:

$P =2 \frac{2}{5} + 3 \frac{1}{5} + 4 = 9 \frac{3}{5}$

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Example Question #72 : Plane Geometry

Right triangle

Figure NOT drawn to scale

$\bigtriangleup ABC$ is a right triangle with altitude $\overline{BX}$ . Give the ratio of the area of $\bigtriangleup BXC$ to that of $\bigtriangleup AXB$ .

Possible Answers:

16:9

25:9

25:16

4:3

Correct answer:

16:9

Explanation:

The altitude of a right triangle from the vertex of its right angle - which, here, is $\overline{BX}$ - divides the triangle into two triangles similar to each other. The ratio of the hypotenuse of the white triangle to that of the gray triangle (which are corresponding sides) is

$\frac{BC}{AB} = \frac{40}{30} =\frac{4}{3}$ ,

making this the similarity ratio. The ratio of the areas of two similar triangles is the square of their similarity ratio, which here is

$\left ( \frac{4}{3} \right )^{2} = \frac{4^{2}}{3^{2}} = \frac{16}{9}$ , or 16: 9 .

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Example Question #4 : How To Find The Area Of A Right Triangle

Find the area of a right triangle with a base of 7cm and a height of 20cm.

Possible Answers:

$70\text{cm}^2$

$110\text{cm}^2$

$120\text{cm}^2$

$140\text{cm}^2$

$80\text{cm}^2$

Correct answer:

$70\text{cm}^2$

Explanation:

To find the area of a right triangle, we will use the following formula:

$A = \frac{1}{2} \cdot b \cdot h$

where b is the base and h is the height of the triangle.

We know the base of the triangle is 7cm. We know the height of the triangle is 20cm. Knowing this, we can substitute into the formula. We get

$A = \frac{1}{2} \cdot 7\text{cm} \cdot 20\text{cm}$

$A = 7\text{cm} \cdot 10\text{cm}$

$A = 70\text{cm}^2$

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Example Question #81 : Geometry

Similar

NOTE: Figures NOT drawn to scale.

Refer to the above two triangles.

$\Delta ABC \sim \Delta DEF ,AB = 25, DE = 20, AC = 20$

What is ?

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

Corresponding sides of similar triangle are proportional, so if

$\Delta ABC \sim \Delta DEF$ , then

$\frac{DF}{AC} = \frac{DE}{AB}$

Substitute the known sidelengths, then solve for :

$\frac{DF}{20} = \frac{20}{25}$

$\frac{DF}{20}\cdot 20= \frac{20}{25}\cdot 20$

$DF = \frac{400}{25}$

DF =16

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Example Question #31 : Triangles

$\Delta ABC \sim \Delta DEF$

AB = 15, BC = 21, AC = 12, DF = 36

What is the perimeter of $\Delta DEF$ ?

Possible Answers:

150

136

128

144

It cannot be determined from the information given.

Correct answer:

144

Explanation:

By definition, since, $\Delta ABC \sim \Delta DEF$ , side lengths are in proportion.

So,

$\frac{DE}{AB}= \frac{DF}{AC}$

$\frac{DE}{15}= \frac{36}{12} = 3$

$\frac{DE}{15} \cdot 15 = 3\cdot 15$

DE = 45

$\frac{EF}{BC}= \frac{DF}{AC}$

$\frac{EF}{21}= \frac{36}{12} = 3$

$\frac{EF}{21} \cdot 21 = 3\cdot 21$

EF = 63

The perimeter of $\Delta DEF$ is

DE + EF + DF = 45 + 63 + 36 = 144 .

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Example Question #83 : Geometry

$\Delta ABC \sim \Delta DEF$

AB = 15, BC = 19, DE = 45, DF = 66

What is ?

Possible Answers:

It is impossible to tell from the information given.

AC = 57

AC = 22

AC = 15

AC = 198

Correct answer:

AC = 22

Explanation:

By definition, since $\Delta ABC \sim \Delta DEF$ , all side lengths are in proportion.

$\frac{AC}{DF} = \frac{AB}{DE}$

$\frac{AC}{66} = \frac{15}{45}$

$\frac{AC}{66} = \frac{1}{3}$

$AC = \frac{1}{3} \cdot 66 = 22$

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ISEE Upper Level Math : Triangles

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

All ISEE Upper Level Math Resources

Example Questions

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

Example Question #31 : Triangles

Example Question #71 : Isee Upper Level (Grades 9 12) Mathematics Achievement

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

Example Question #71 : Plane Geometry

Example Question #72 : Plane Geometry

Example Question #4 : How To Find The Area Of A Right Triangle

Example Question #81 : Geometry

Example Question #31 : Triangles

Example Question #83 : Geometry

All ISEE Upper Level Math Resources

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