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Geometry Word Problems Study Guide (page 3)

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Updated on Aug 24, 2011

Example

Triangles ABC and DEF are similar. If the length of side AB is 12 feet, the length of side DE is 30 feet, and the length of BC is 8 feet, what is the length of side EF?

Since this is a similar triangle word problem, we'll solve it by drawing a picture. Start by drawing two triangles. It's not important how they look, but it is important how we label them. We must be sure to match angle A with angle D, angle B with angle E, and angle C with angle F:

Triangles: Subject Review

Next, label AB 12 feet, DE 30 feet, and BC 8 feet:

Triangles: Subject Review

We can see now that sides AB and DE are corresponding and sides BC and EF are corresponding. Since the triangles are similar, the ratio of side AB to side DE is equal to the ratio of side BC to side EF: . Cross multiply and divide: 12EF = 240, EF = 20 feet.

We can identify a problem that involves a right triangle if we see the word hypotenuse. The word problem may also state that the triangle is a right triangle, or it may simply state that the triangle contains a 90° or right angle. If a problem asks for the missing side of a right triangle, you will likely need to use the Pythagorean theorem.

Example

The legs of triangle GHI form a right angle. If the length of one leg is 15 feet and the length of the other leg is 20 feet, what is the length of the hypotenuse?

Read the entire word problem.

We are given the lengths of the legs of a triangle that has a right angle.

Identify the question being asked.

We are looking for the length of the hypotenuse.

Underline the keywords and words that indicate formulas.

This problem contains many keywords. The legs of the triangle form a right angle, which means that this is a right triangle. The word hypotenuse also tells us that this is a right triangle, and in order to find the length of the hypotenuse, we will have to use the Pythagorean theorem: a2 + b2 = c2.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

We will need to square the legs and add those squares. Then, we will need to take the square root of that sum.

Write number sentences for each operation.

First, square the legs and add them:

(15)2 + (20)2

Solve the number sentences and decide which answer is reasonable.

(15)2 + (20)2 = 225 + 400 = 625

Write number sentences for each operation.

The answer 625 is the square of the hypotenuse, so to find the length of the hypotenuse, we must take the square root of 625:

625

Solve the number sentences and decide which answer is reasonable.

625 = 25 feet

The hypotenuse of triangle GHI is 25 feet.

Check your work.

The square of the hypotenuse should equal the sum of the squares of the legs: (15)2 + (20)2 = (25)2, 225 + 400 = 625, 625 = 625.

Inside Track

Only a right triangle has a hypotenuse, so even if you are not told that a triangle is right, if you are asked to find the length of the hypotenuse, you are working with a right triangle, and you will need to use the Pythagorean theorem.

Polygons and Quadrilaterals: Subject Review

The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon. If a polygon has ten sides, then the sum of its interior angles is 180(10 – 2) = 180(8) = 1,440°.

There are six major types of quadrilaterals: parallelograms, rhombi, rectangles, squares, trapezoids, and isosceles trapezoids. By analyzing the properties of a quadrilateral, we can determine if the quadrilateral is a parallelogram, rhombus, rectangle, square, trapezoid, isosceles triangle—or just a quadrilateral.

Fuel for Thought

An interior angle is an angle found within a closed figure.

A polygon is a closed figure made up of line segments.

A quadrilateral is a polygon with four sides.

A parallelogram is a quadrilateral whose opposite sides are parallel and congruent. Its opposite angles are also congruent, consecutive angles are supplementary, and its diagonals bisect each other.

A rectangle is a parallelogram with four congruent, 90° angles and congruent diagonals.

A rhombus is a parallelogram with four congruent sides and perpendicular diagonals that bisect their angles.

A square has all of the properties of a rhombus and a rectangle: four congruent sides, four congruent, 90° angles, two pairs of parallel sides, consecutive angles that are supplementary, and diagonals that are congruent, perpendicular, bisect each other, and bisect their angles.

A trapezoid is a quadrilateral with at least one pair of parallel sides.

An isosceles trapezoid is a trapezoid with a pair of congruent sides, angles, and diagonals.

The formula for the area of any parallelogram is A = bh, where A is the area, b is the base of the parallelogram, and h is the height of the parallelogram. If the base of a parallelogram is 10 centimeters and the height is 40 centimeters, then the area is (10)(40) = 400 centimeters2.

This formula can also be used to find the area of a rectangle, since a rectangle is a type of parallelogram, but the area of a rectangle is usually described as A = lw, where A is the area, l is the length of the rectangle, and w is the width of the rectangle. If the length of a rectangle is 1.2 millimeters and the width is 2.5 millimeters, then the area is (1.2)(2.5) = 3 millimeters2.

The area of a square could be found with either formula, since a square is both a parallelogram and a rectangle, but it is easier to describe the area of a square as A = s2, where s is the length of one side of the square, since all four sides of a square are congruent. If one side of a square is 0.17 meters, then the area of the square is (0.17)2 = 0.0289 meters2.

The area of a trapezoid is A = (b1+ b2)h, where A is the area, b1 and b2 are the bases of the trapezoid, and h is the height of the trapezoid. If the lengths of the bases of a trapezoid are 3 feet and 9 feet, respectively, and the height is 10 feet, then the area of the trapezoid is (3 + 9)10 = (12)(10) = (6)(10) = 60 feet2.

The perimeter of any quadrilateral is the sum of the lengths of each of its four sides, but we also have separate formulas for rectangles and squares. The perimeter of a rectangle can be described as P = 2l + 2w, since the two lengths of the rectangle are the same, as are the two widths. Since all four sides of a square measure the same, the perimeter of a square can be described as P = 4s. Just multiply the length of one side by 4. If one side of a square is 5.4 centimeters, then the perimeter of the square is (5.4)(4) = 21.6 centimeters.

Just as with triangles, quadrilateral word problems about area or perimeter usually contain the word area or the word perimeter. Phrases such as distance around can describe perimeter, and phrases such as surface covered can indicate area.

Example

What is the distance around a rectangle that has a length of 2 yards and a width of 8.9 yards?

Read the entire word problem.

We are given the length and width of a rectangle.

Identify the question being asked.

We are looking for the distance around the rectangle.

Underline the keywords and words that indicate formulas.

The words distance around indicate that we are looking for the perimeter of the rectangle, which is equal to 2l + 2w.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

We will need to multiply both the length and the width by 2 and then add those products.

Write number sentences for each operation.

2(2) + 2(8.9)

Solve the number sentences and decide which answer is reasonable.

2(2) + 2(8.9) = 4 + 17.8 = 21.8 yards

Check your work.

Subtract the length of each side from the perimeter. Since the perimeter is the total distance around the rectangle, we should be left with zero after subtracting every length: 21.8 – 2 – 2 – 8.9 – 8.9 = 0

Word problems can combine more than one formula. For instance, you may use the perimeter of a square to find the length of one side of that square, and then use the length of the side to find the area of the square.

Example

If the perimeter of a square is 44 inches, what is the area of the square?

Read the entire word problem.

We are given the perimeter of a square.

Identify the question being asked.

We are looking for the area of the square.

Underline the keywords and words that indicate formulas.

The words perimeter and area are keywords.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

The perimeter of a square is equal to 4s, so we will need to divide 44 by 4 to find the length of one side of the square. The area of a square is equal to s2, so we will need to square our quotient.

Write number sentences for each operation.

First, find the length of one side of the square:

Solve the number sentences and decide which answer is reasonable.

= 11 inches

Write number sentences for each operation.

One side of the square is 11 inches, which means that the area of the square is equal to 112.

Solve the number sentences and decide which answer is reasonable.

112 = 121 square inches

Check your work.

Take the square root of the area to find the length of one side of the square. Then, multiply that by 4 to find the perimeter of the square, which should equal 44 inches: √121 = 11, (4)(11) = 44 inches.

Caution

When working with a word problem that involves both area and perimeter, be sure watch your units carefully. Perimeter is found in linear units, and area is found in square units.

Pace Yourself

Using a ruler, a pencil, and paper, draw a rectangle. What is the area and perimeter of your rectangle? Now, divide your rectangle into two triangles. What is the area and the perimeter of each triangle?

Circles: Subject Review

The formula for area of a circle is A = πr2, where A is the area of the circle and r is the radius of the circle. If a circle has a radius of 10 feet, then its area is π(10)2 = 100π feet2.

Fuel for Thought

A circle is a set of connected points that are all the same distance from a single point, which is the center of the circle. A circle is a closed figure, but not a polygon.

A radius is the distance from the center of a circle to a point on the circle.

A diameter is the distance from one side of a circle to the other through the center of the circle.

The circumference of a circle is the distance around the outside of a circle.

The diameter of a circle is twice the radius, which means that d = 2r and r = d, where d is the diameter and r is the radius. If the diameter of a circle is 27 feet, then the radius of the circle is (2) = 13.5 feet. If the radius of another circle is 2.8 centimeters, then the diameter of that circle is 2(2.8) = 5.6 centimeters.

The circumference of a circle is equal to 2πr, or πd. If the diameter of a circle is 27 feet, then the circumference of that circle is 27π feet. If the circumference of a circle is 10π inches, then the diameter of the circle is 10 inches and the radius is 5 inches.

Circumference sounds a lot like perimeter, and it may appear in word problems the same way. If a word problem asks you for the distance around a circle, it's asking you for the circumference of the circle. The keywords radius and diameter will also indicate that you are working with a circle, although the word circle almost always appears in any word problem that involves a circle.

Inside Track

Since we have two formulas for the circumference of a circle, one that uses the radius and one that uses the diameter, choose which formula to use based on the information given to you in a problem. If you are given the radius, use C = 2πr, and if you are given the diameter, use C= πd. You may save yourself the step of finding the radius from the diameter, or vice versa.

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