### All High School Math Resources

## Example Questions

### Example Question #141 : Plane Geometry

You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?

**Possible Answers:**

30

23

25

35

33

**Correct answer:**

30

Similar triangles are proportional.

Base_{1} / Height_{1} = Base_{2} / Height_{2}

6 / 9 = 20 / Height_{2}

Cross multiply and solve for Height_{2}

6 / 9 = 20 / Height_{2}

6 * Height_{2}= 20 * 9

Height_{2}= 30

### Example Question #1 : Right Triangles

In the figure above, line segments *DC* and *AB* are parallel. What is the perimeter of quadrilateral *ABCD*?

**Possible Answers:**

75

85

95

80

90

**Correct answer:**

85

Because *DC* and *AB* are parallel, this means that angles *CDB* and *ABD* are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as *CDB* and *ABD*) are congruent.

Now, we can show that triangles *ABD* and *BDC* are similar. Both *ABD* and *BDC* are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles *CDB* and *ABD* are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles *ABD* and *BDC* are similar triangles.

We can use the similarity between triangles *ABD* and *BDC* to find the lengths of *BC* and *CD*. The length of *BC* is proportional to the length of *AD*, and the length of *CD* is proportional to the length of *DB*, because these sides correspond.

We don’t know the length of *DB*, but we can find it using the Pythagorean Theorem. Let *a*, *b*, and *c* represent the lengths of *AD*, *AB*, and *BD* respectively. According to the Pythagorean Theorem:

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

15^{2} + 20^{2 }= *c*^{2}

625 = *c*^{2}

*c* = 25

The length of *BD* is 25.

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of *AB*, *BC*, *CD*, and *DA*.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85.

### Example Question #142 : Plane Geometry

A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?

**Possible Answers:**

81 square units

108 square units

36 square units

None of the answers are correct

54 square units

**Correct answer:**

54 square units

The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by A_{new} = (SF)^{2} x (A_{old}) =

3^{2} x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).

NOTE: For a volume problem: V_{new} = (SF)^{3} x (V_{old}).

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

Solve for (rounded to the nearest tenth). Figure not drawn to scale.

**Possible Answers:**

**Correct answer:**

We will use the Pythagorean Theorem to solve for the missing side length.

### Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

A square boxing ring has a perimeter of feet. When the two boxers are sitting in opposite corners between rounds, how far apart are they?

**Possible Answers:**

feet

feet

feet

feet

feet

**Correct answer:**

feet

Since the perimeter of the ring is feet and the ring is a square, solve for the length of a single side of the ring by dividing by .

The distance between the two boxers in opposing corners is a straight line from any one corner to the other. That straight line forms the hypotenuse of a right triangle whose other two sides are each feet long (since they are each the sides of the square).

Solving for the length of the hypotenuse of this right triangle with the pythagorean theorem provides the distance between the two boxers when they are in opposite corners.

### Example Question #1 : Right Triangles

Given a right triangle with a leg length of 6 and a hypotenuse length of 10, find the length of the other leg, *x*.

**Possible Answers:**

16

4

8

64

**Correct answer:**

8

Using Pythagorean Theorem, we can solve for the length of leg *x*:

*x*^{2} + 6^{2} = 10^{2}

Now we solve for *x*:

*x*^{2} + 36 = 100

*x*^{2} = 100 – 36

*x*^{2} = 64

*x* = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

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

In a right triangle a hypotenuse has a length of 8 and leg has a length of 7. What is the length of the third side to the nearest tenth?

**Possible Answers:**

**Correct answer:**3.9

Using the pythagorean theorem, 8^{2}=7^{2}+x^{2}. Solving for x yields the square root of 15, which is 3.9^{}

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

Given a right triangle with a leg length of 2 and a hypotenuse length of √8, find the length of the other leg, *x*.

**Possible Answers:**

10

6

4

√8

2

**Correct answer:**

2

Using Pythagorean Theorem, we can solve for the length of leg *x*:

*x*^{2} + 2^{2} = (√8)^{2 }= 8

Now we solve for *x*:

*x*^{2} + 4 = 8

*x*^{2} = 8 – 4

*x*^{2} = 4

*x* = 2

### Example Question #1 : Right Triangles

The legs of a right triangle are and . Rounded to the nearest whole number, what is the length of the hypotenuse?

**Possible Answers:**

**Correct answer:**

Use the Pythagorean Theorem. The sum of both legs squared equals the hypotenuse squared.

### Example Question #7 : How To Find The Length Of The Side Of A Right Triangle

**Possible Answers:**

**Correct answer:**

Use the Pythagorean theorem: .

We know the length of one side and the hypotenuse.

Now we can solve for the missing side.