Geometry

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Find the perimeter of the triangle above.

Note: Figure not drawn to scale.

None of these answers are correct.

Explanation

The perimeter of a shape is the length around the shape. In order to find the perimeter of a triangle, add the lengths of the sides: .

Because the lengths are in inches, the answer must be in inches as well.

In Parallelogram , and . Which of the following is greater?

(A)

(B)

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Explanation

In Parallelogram , $\overline{AB}$ and $\overline{BC }$ are adjoining sides; there is no specific rule for the relationship between their lengths. Therefore, no conclusion can be drawn of and , and no conclusion can be drawn of the relationship between and .

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid . Give the ratio of the area of Trapezoid to that of Trapezoid .

33 to 19

10 to 3

13 to 6

20 to 13

Explanation

Midsegment $\overline{XY}$ divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

$XY = \frac{1}{2} (12 + 40) = \frac{1}{2} \cdot 52 = 26$ .

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+26 ) = \frac{1}{2} \cdot 38 \cdot h = 19 h$

The area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (XY +AP) = \frac{1}{2} \cdot h \cdot ( 26+40 ) = \frac{1}{2} \cdot 66 \cdot h = 33 h$

The ratio of the areas is

$\frac{33 h}{19h} = \frac{33}{19}$ , or 33 to 19.

Given Trapezoid , where $\overline{TR} \parallel \overline{PA}$ . Also, $m \angle A > m\angle P$

Which is the greater quantity?

(a) $m\angle T$

(b) $m \angle R$

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

$\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ . Therefore,

$m \angle T + m \angle P = 180$ , or $m \angle P = 180 - m \angle T$

$m \angle R + m \angle A = 180$ , or $m \angle A = 180 - m \angle R$

Substitute:

$m \angle A > m\angle P$

$180 - m \angle R > 180 - m\angle T$

$180 - m \angle R -180 > 180 - m\angle T-180$

$- m \angle R > - m\angle T$

$- (- m \angle R )<-\left ( - m\angle T \right )$

$m \angle R < m\angle T$

(a) is the greater quantity

Trapezoid

The above diagram depicts trapezoid . Which is the greater quantity?

(a) $m \angle T + m \angle P$

(b) $m \angle R + m \angle A$

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

Explanation

$\overline{TR} \parallel \overline{PA}$ ; $\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ .

Therefore, $m \angle T + m \angle P = 180= m \angle R + m \angle A$ , making the two quantities equal.

Find the length of an edge of the following cube:

Length_of_edge

The volume of the cube is $27\ m^3$ .

$3\ m$

$6\ m$

$9\ m$

$12\ m$

$15\ m$

Explanation

The formula for the volume of a cube is

where is the length of the edge of a cube.

Plugging in our values, we get:

$27\ m^3 = (e)^3$

$e = 3\ m$

A cube has a side length of meters. What is the volume of the cube?

$8\ \textup{m}^{3}$

$6\ \textup{m}^{3}$

$4\ \textup{m}^{3}$

$2\ \textup{m}^{3}$

$16\ \textup{m}^{3}$

Explanation

The formula for the volume of a cube is:

$Volume = (side)^{3}$

Since the length of one side is meters, the volume of the cube is:

$2^{3 } = 8$ meters cubed.

Find the volume of the following triangular prism:

Triangular_prism

$108\sqrt{3}m^3$

$98\sqrt{3}m^3$

$88\sqrt{3}m^3$

$118\sqrt{3}m^3$

$128\sqrt{3}m^3$

Explanation

The formula for the volume of an equilateral, triangular prism is:

$V = \left(\frac{s^2\sqrt{3}}{4}\right)(h)$

Where is the length of the triangle side and is the length of the height.

Plugging in our values, we get:

$V = \left(\frac{(6m)^2\sqrt{3}}{4}\right)(12m)$

$V=108\sqrt{3}m^3$

The perimeter of a square is 48. What is the length of its diagonal?

$12\sqrt{2}$

$\sqrt{12}$

$24\sqrt{2}$

$48\sqrt{2}$

$2\sqrt{12}$

Explanation

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

$\sqrt{144+144}=\sqrt{hypotenuse^2}$

$hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}$

Example circle

Find the diameter, circumference and area of the circle above.

Diameter= 6 ft

Circumference=18.84 ft

Area= 28.27 ft2

Diameter= 6 ft

Circumference= 19 ft

Area= 30 ft2

Diameter= 9ft

Circumference=37.68 ft

Area= 28 ft2

Diameter= 3ft

Circumference=37.68 ft

Area= 28.7 ft2

Diameter= 6ft

Circumference=37.68 ft

Area= 28.27 ft2

Explanation

To find the diameter, you must know that radius is half the diameter (or the diameter is 2 times the radius.

$\small Diameter= 2(radius)$

$\small \small 2(3)= 6$

The diameter is 6ft.

To find the circumference, you must multiply the diameter (6ft) by pi.

$\small Circumference=\pi (diameter)$

$\small \small \pi(6)= 18.84$

The circumference is 18.84 ft.

To find the surface area, you must aquare the radius (3ft) and multiply by pi.

$\small Surface Area=\pi (radius^{2})$

$\small \pi(3{^2})=\pi(9)=28.27$

The surface area is 28.27 ft2.

The diameter is 6ft, the circumference is 18.84 ft, and the surface area is 28.27 ft2.

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