Triangles - ISEE Upper Level Quantitative Reasoning

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Question

A right triangle has a hypotenuse of 10 and a side of 6. What is the missing side?

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Answer

To find the missing side, use the Pythagorean Theorem . Plug in (remember c is always the hypotenuse!) so that . Simplify and you get Subtract 36 from both sides so that you get Take the square root of both sides. B is 8.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

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

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Answer

First, find .

Since $\overline{DF }$ is an altitude of right $\Delta ADB$ to its hypotenuse,

$\frac{FB}{FD}$= $\frac{FD}{AF}$

$\frac{FB}{8}$= $\frac{8}{6}$

$FB= $\frac{8}{6}$ \cdot 8 = 10 $\frac{2}{3}$$

$AB = AF + FB = 6 + 10 $\frac{2}{3}$ = 16 $\frac{2}{3}$$

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

$\frac{BC}{FD}$ = $\frac{AB}{AF}$

$$\frac{BC}{8}$ = $\frac{16 \frac{2}{3}$}{6}$

$BC = 16 $\frac{2}{3}$\cdot 8 \div 6 = 22$\frac{2}{9}$$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

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

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Answer

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$

$\Delta DEC \sim \Delta ABC$ 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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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

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Answer

By the Pythagorean Theorem,

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

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following quadratic equations would yield the value of as a solution?

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Answer

By the Pythagorean Theorem,

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Question

Refer to the above diagram. Which of the following quadratic equations would yield the value of as a solution?

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Answer

By the Pythagorean Theorem,

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Question

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

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Answer

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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Question

Refer to the above figure. The shaded region is a semicircle with area $16 \pi$ . Give the perimeter of $\bigtriangleup ABC$ .

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Answer

Given the radius of a semicircle, its area can be calculated using the formula

$A = $\frac{1}{2}$ \pi r ^{2}$ .

Substituting $A = 16 \pi$ :

$$\frac{1}{2}$ \pi r ^{2} = 16 \pi$

$$\frac{2}{\pi}$ \cdot $\frac{1}{2}$ \pi r ^{2} = $\frac{2}{\pi}$ \cdot 16 \pi$

$r ^{2} = 32$

$r = $\sqrt{32}$ = $\sqrt{16}$ \cdot $\sqrt{2}$ = 4$\sqrt{2}$$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4$\sqrt{2}$ = 8 $\sqrt{2}$$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 $\sqrt{2}$$ . Its perimeter is three times this, or $3 \cdot 8 $\sqrt{2}$ = 24 $\sqrt{2}$$

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Question

Find the perimeter of an equilateral triangle with a base of length 8cm.

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Answer

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

where a, b, and c are the lengths of the sides of the triangle

Now, we know the base of the triangle is 8cm. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 8cm.

Knowing this, we can substitute into the formula. We get

$P = 8$\text{cm}$ + 8$\text{cm}$ + 8$\text{cm}$$

$P = 24$\text{cm}$$

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Question

Find the perimeter of an equilateral triangle with a base of 21in.

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Answer

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

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 21in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 21in.

Knowing this, we can substitute into the formula. We get

$P = 21$\text{in}$ +21$\text{in}$ +21$\text{in}$$

$P = 63$\text{in}$$

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Question

Find the perimeter of an equilateral triangle with a base of 23in.

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Answer

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

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 23in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 23in.

Knowing this, we can substitute into the formula. We get

$P = 23$\text{in}$ +23$\text{in}$ +23$\text{in}$$

$P = 69$\text{in}$$

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Question

Find the perimeter of an equilateral triangle with a base of 22in.

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Answer

An equilateral triangle has 3 equal sides. So, we will use the following formula:

where a is the length of one side of the triangle.

Now, we know the base of the triangle is 22in. Because all sides are equal, all sides are 22in. So, we can substitute into the formula:

$P = 3 \cdot 22$\text{in}$$

$P = 66$\text{in}$$

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Question

Find the perimeter of an equilateral triangle with a base of 19in.

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Answer

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

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle is 19in. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 19in. So, we get

$P = 19$\text{in}$ + 19$\text{in}$ + 19$\text{in}$$

$P = 57$\text{in}$$

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Question

Find the perimeter of an equilateral triangle with a base of 14in.

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Answer

An equilateral triangle has 3 equal sides. To find the perimeter of an equilateral triangle, we will use the following formula:

where a is the length of one side of the triangle.

Now, we know the base of the equilateral triangle is 14in. So, we get

$P = 3 \cdot 14$\text{in}$$

$P = 42$\text{in}$$

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Question

Which of the following could be the lengths of the three sides of a scalene triangle?

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Answer

A scalene triangle, by definition, has three sides of different lengths. We can identify the scalene triangle by converting measures to the same unit. We will convert to inches in this case.

$5 \textrm{ ft, }60 \textrm{ in, }4\textrm{ ft}$

5 feet = $5 \times 12 = 60$ inches. This triangle can be eliminated, since two sides have length 60 inches.

$7 \textrm{ ft, }84 \textrm{ in, }72\textrm{ in}$

7 feet = $7 \times 12 = 84$ inches. This triangle can be eliminated, since two sides have length 84 inches.

$3 \textrm{ ft, }36 \textrm{ in, }1\textrm{ yd}$

These three measures are equal. This is an equilateral triangle, and it can be eliminated.

$5 \textrm{ ft, }66 \textrm{ in, }5 $\frac{1}{2}$\textrm{ ft}$

$5 $\frac{1}{2}$$ feet = $5 $\frac{1}{2}$ \times 12 = 66$ inches. This triangle can be eliminated, since two sides have length 66 inches.

$6 \textrm{ ft, }84 \textrm{ in, }5\textrm{ ft}$

5 feet = 60 inches.

6 feet = $6\times 12 = 72$ inches.

The sides of this triangle measure 60, 72, and 84 inches, so the triangle is scalene.

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Question

For an equilateral triangle, Side A measures and Side B measures . What is the length of Side A?

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Answer

First you need to recognize that for an equilateral triangle, all 3 sides have equal lengths.

This means you can set the two values for Side A and Side B equal to one another, since they measure the same length, to solve for .

You now know that , but this is not your answer. The question asked for the length of Side A, so you need to plug 3 into that equation.

So the length of Side A (and Side B for that matter) is 8.

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Question

One angle of an equilateral triangle is 60 degrees. One side of that triangle is 12 centimeters. What are the measures of the two other angles and two other sides?

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Answer

An equilateral triangle is one in which all three sides are congruent. It also has the property that all three interior angles are equal. In other words, all three angles of an equilateral triangle are always 60°. Since all sides are congruent, the other two sides both measure 12 centimeters.

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Question

If an equilateral triangle has a perimeter of 18in, what is the length of one side?

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Answer

To find the perimeter of an equilateral triangle, we will use the following formula:

where a is the length of any side of the triangle. Because an equilateral triangle has 3 equal sides, we can use any side in the formula. To find the length of one side of the triangle, we will solve for a.

Now, we know the perimeter of the equilateral triangle is 18in. So, we will substitute.

$18$\text{in}$ = 3a$

$\frac{18\text{in}$}{3} = $\frac{3a}{3}$

$6$\text{in}$ = a$

$a=6$\text{in}$$

Therefore, the length of one side of the equilateral triangle is 6in.

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Question

An equilateral triangle has a perimeter of 39in. Find the length of one side.

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Answer

An equilateral triangle has 3 equal sides. So, to find the length of one side, we will use what we know. We know the perimeter of the equilateral triangle is 39in. So, we will look at the formula for perimeter. We get

where a is the length of one side of the triangle. So, to find the length of one side, we will solve for a. Now, as stated before, we know the perimeter of the triangle is 39in. So, we will substitute and solve for a. We get

$39$\text{in}$ = 3a$

$\frac{39\text{in}$}{3} = $\frac{3a}{3}$

$13$\text{in}$ = a$

$a = 13$\text{in}$$

Therefore, the length of one side of the equilateral triangle is 13in.

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Question

Refer to the above figure. The shaded region is a semicircle with area $16 \pi$ . Give the perimeter of $\bigtriangleup ABC$ .

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Answer

Given the radius of a semicircle, its area can be calculated using the formula

$A = $\frac{1}{2}$ \pi r ^{2}$ .

Substituting $A = 16 \pi$ :

$$\frac{1}{2}$ \pi r ^{2} = 16 \pi$

$$\frac{2}{\pi}$ \cdot $\frac{1}{2}$ \pi r ^{2} = $\frac{2}{\pi}$ \cdot 16 \pi$

$r ^{2} = 32$

$r = $\sqrt{32}$ = $\sqrt{16}$ \cdot $\sqrt{2}$ = 4$\sqrt{2}$$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4$\sqrt{2}$ = 8 $\sqrt{2}$$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 $\sqrt{2}$$ . Its perimeter is three times this, or $3 \cdot 8 $\sqrt{2}$ = 24 $\sqrt{2}$$

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