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Intermediate Geometry : Plane Geometry

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Example Questions

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

Given: Quadrilateral ABCD with diagonal $\overline{AC}$ ; $\bigtriangleup ABC \cong \bigtriangleup CDA$ .

True or false: From the information given, it follows that Quadrilateral ABCD is a parallelogram.

Possible Answers:

True

False

Correct answer:

True

Explanation:

Corresponding parts of congruent triangles are, by definition, congruent. Thus, from the statement $\bigtriangleup ABC \cong \bigtriangleup CDA$ , it follows that:

$\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$

Quadrilateral ABCD therefore has two sets of congruent opposite sides. This is a sufficient condition for the quadrilateral to be a parallelogram.

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Example Question #212 : Quadrilaterals

Quadrilateral ABCD is both a rhombus and a rectangle.

True or false: Quadrilateral ABCD must be a square.

Possible Answers:

True

False

Correct answer:

True

Explanation:

A rhombus is defined to be a parallelogram with four congruent sides; a rectangle is defined to be a parallelogram with four right angles.

A square is defined to be a parallelogram with four congruent sides and four right angles. If a parallelogram is both a rhombus and a rectangle, then it fits both characteristics and is therefore a square.

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

The perimeter of a square is $24\; in$ . If the sides of the square are reduced by a factor of two, what is the area of the new square?

Possible Answers:

$9\; in^{2}$

$45\; in^{2}$

$27\; in^{2}$

$36\; in^{2}$

$18\; in^{2}$

Correct answer:

$9\; in^{2}$

Explanation:

The perimeter of a square is geven by P=4s and the area of a square is given by $A=s^{2}$ .

Thus $P=4s=24\: in$

so $s=6\; in$ .

The original side is reduced by a factor of which results in a new side of $3\; in$ . The area of the new square is geven by

$A=s^{2}=3^{2}=9\; in^{2}$

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Example Question #2 : How To Find The Area Of A Parallelogram

Find the area of the box in square inches:

Geo_box

Possible Answers:

84xy

x + y + 19

2x + 2y + 38

xy + 7x + 12y +84

12x +7y

Correct answer:

xy + 7x + 12y +84

Explanation:

The answer is xy + 7x + 12y +84 . You can find the area of this box by multiplying the length by its width:

Foil (12 + x) (7 + y ) = xy + 7x + 12y +84

If you chose 2x + 2y + 38 , you added the sides to get the perimeter.

Just remember, the width is 12 added to . Not 12 times the side of .

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Example Question #2 : How To Find The Area Of A Parallelogram

If all angles are right angles in the figure below, what is the total area of the figure?

Box1

Possible Answers:

$256\ in^{2}$

$152\ in^{2}$

$184\ in^{2}$

$308\ in^{2}$

$164\ in^{2}$

Correct answer:

$164\ in^{2}$

Explanation:

The answer is $164\ in^{2}$ .

To get the area, you would need to find the missing sides. Since the horizontal top is 8+8=16 inches total, the bottom must be inches total. Thus you would subtract 16-6 to get for the missing bottom horizontal side.

The vertical right side is 4+12 = 16 inches total. Then you would subtract 16-10 to get for the missing vertical side.

Then you would have to split the figure into 3 different boxes and find the area of each one. Then you would add the area of each box

$(32+60+72)=164\ in ^{2}$

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

What is the area of a parallelogram if the base is $1.5\:cm$ , and the height is $1.0\:cm$ ?

Possible Answers:

$0.25\:cm$

$0.75\:cm$

$3\:cm$

$1.5\sqrt2\:cm$

$1.5\:cm$

Correct answer:

$1.5\:cm$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$

So, we are given all of the information that we need to solve this problem. Substitute the provided values into the equation and simplify:

$A= (1.5\:cm)(1\:cm) =1.5\:cm^2$

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Example Question #5 : How To Find The Area Of A Parallelogram

What is the area of a parallelogram if the base is $2x\:cm$ and the height is $x^2\:cm$ ?

Possible Answers:

$2x^3\:cm^2$

$x^3\:cm^2$

$2x+x^2\:cm^2$

$2\sqrt2x^3\:cm^2$

$2x^2\:cm^2$

Correct answer:

$2x^3\:cm^2$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$

Substitute in the provided values and simplify to calculate the correct answer:

$A= (2x\:cm)(x^2\:cm) =2x^3\:cm^2$

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Example Question #6 : How To Find The Area Of A Parallelogram

Find the area of a parallelogram if the base is (x-2) and the height is (x+4) .

Possible Answers:

x^2+2x-8

4x+4

2x+2

x^2-2x-8

x^2-8

Correct answer:

x^2+2x-8

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$

Substitute the values provided in the question:

A= (x-2)(x+4)

Use the FOIL method to simplify:

A=(x)(x)+(x)(4) +(-2)(x)+(-2)(4)

A= x^2+4x-2x-8

A=x^2+2x-8

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Example Question #6 : How To Find The Area Of A Parallelogram

What is the area of a parallelogram if the base is $\sqrt2\:cm$ and the height is $\sqrt3\:cm$ ?

Possible Answers:

$\frac{\sqrt5}{2}\:cm^2$

$\frac{\sqrt6}{2}\:cm^2$

$\sqrt6\:cm^2$

$\sqrt5\:cm^2$

$\sqrt2 + \sqrt3\:cm^2$

Correct answer:

$\sqrt6\:cm^2$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$

Substitute the values provided in the problem and solve to calculate the correct answer:

$A= (\sqrt2\:cm)(\sqrt3\:cm) =\sqrt6\:cm^2$

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

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

Correct answer:

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=2 \times 6$

$\text{Area of Rectangle}=12$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{2}{2}$

$\text{height}=1$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=3 \times 1$

$\text{Area of Parallelogram}=3$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=12-3$

Solve.

$\text{Area of Shaded Region}=9$

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #212 : Quadrilaterals

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

Example Question #2 : How To Find The Area Of A Parallelogram

Example Question #2 : How To Find The Area Of A Parallelogram

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

Example Question #5 : How To Find The Area Of A Parallelogram

Example Question #6 : How To Find The Area Of A Parallelogram

Example Question #6 : How To Find The Area Of A Parallelogram

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

All Intermediate Geometry Resources

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