Plane Geometry

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Find the length of the arc if the radius of a circle is and the measure of the central angle is degrees.

$\frac{11}{3}\pi$

$4\pi$

$\frac{13}{3}\pi$

$5\pi$

Explanation

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{55}{360}\times 2\pi \times 12$

Solve.

$\text{Length of Arc}=\frac{11}{3}\pi$

In the figure below, $\text{chord AB}\parallel\text{chord CD}$ . If $\text{Arc AC}$ is degrees, in degrees, what is the measure of $\text{Arc BD}$ ?

The measurement of $\text{Arc BD}$ cannot be determined with the information given.

Explanation

Recall that when chords are parallel, the arcs that are intercepted are congruent. Thus, $\text{Arc AC}=\text{Arc BD}$ .

Then, $\text{Arc BD}$ must also be degrees.

Given: Rhombuses and .

$m \angle A = 60 ^{\circ }$ and $m \angle F = 120 ^{\circ }$

True, false, or undetermined: Rhombus $ABCD \sim$ Rhombus .

True

False

Undetermined

Explanation

Two figures are similar by definition if all of their corresponding sides are proportional and all of their corresponding angles are congruent.

By definition, a rhombus has four sides that are congruent. If we let $s_{1}$ be the common sidelength of Rhombus and $s_{2}$ be the common sidelength of Rhombus , it can easily be seen that the ratio of the length of each side of the former to that of the latter is the same ratio, namely, $\frac{s_{1}}{s_{2}}$ .

Also, a rhombus being a parallelogram, its opposite angles are congruent, and its consecutive angles are supplementary. Therefore, since $m \angle A = 60 ^{\circ }$ , it follows that $m\angle C = 60 ^{\circ }$ , and $m \angle B = m \angle D = 180 ^{\circ } - 60^{\circ } = 120 ^{\circ }$ . By a similar argument, $m \angle H = m \angle F = 120 ^{\circ }$ and $m \angle E = m \angle G= 180 ^{\circ } - 120 ^{\circ } = 60^{\circ }$ . Therefore,

$\angle A \cong \angle E$

$\angle B \cong \angle F$

$\angle C \cong \angle F$

$\angle D \cong \angle H$

Since all corresponding sides are proportional and all corresponding angles are congruent, it holds that Rhombus $ABCD \sim$ Rhombus .

A rhombus has a side length of $\frac{5}{6}$ foot, what is the length of the perimeter (in inches).

inches

feet

inches

Explanation

To find the perimeter, first convert $\frac{5}{6}$ foot into the equivalent amount of inches. Since, $\frac{5}{6}=\frac{10}{12}$ and $\frac{12}{12}=1foot$ , $\frac{5}{6}$ is equal to inches.

Then apply the formula , where is equal to the length of one side of the rhombus.

Since,

The solution is:

A triangle is placed in a parallelogram so they share a base.

If the height of the triangle is half that of the parallelogram, find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we will need to find the areas of the triangle and of the parallelogram.

First, recall how to find the area of a parallelogram.

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

$\text{Area of Parallelogram}=12\times8=96$

Next, recall how to find the area of a triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

Now, find the height of the triangle.

$\text{Height of Triangle}=\frac{\text{Height of Parallelogram}}{2}$

$\text{Height of Triangle}=\frac{8}{2}=4$

Plug this value in to find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(12 \times 4)=24$

Subtract the two areas to find the area of the shaded region.

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

$\text{Area of Shaded Region}=96-24=72$

In ΔABC: a = 10, c = 15, and B = 50°.

Find b (to the nearest tenth).

11.5

9.7

12.1

11.9

14.3

Explanation

This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = ? a = 10

B = 50° b = ?

C = ? c = 15

Now we can easily see that we do not have any complete pairs (such as A and a, or B and b) but we do have one term from each pair (a, B, and c). This tells us that we should use the Law of Cosines. (We use the Law of Sines when we have one complete pair, such as B and b).

Law of Cosines:
$a^2 = b^2 + c^2 - 2bc\cos(A)$
$b^2 = a^2 + c^2 - 2ac\cos(B)$
$c^2 = a^2 + b^2 - 2ab\cos(C)$

Since we are solving for b, let's use the second version of the Law of Cosines.
This gives us:

$b^2 = 10^2 + 15^2 - 2(10)(15)\cos(50)$

The largest angle in an obtuse scalene triangle is degrees. The second largest angle in the triangle is $\frac{1}{3}$ the measurement of the largest angle. What is the measurement of the smallest angle in the obtuse scalene triangle?

$20^\circ$

$40^\circ$

$30^\circ$

$25^\circ$

$15^\circ$

Explanation

Since this is a scalene triangle, all of the interior angles will have different measures. However, it's fundemental to note that in any triangle the sum of the measurements of the three interior angles must equal degrees.

The largest angle is equal to degrees and second interior angle must equal:

$120\div3=40$

Therefore, the final angle must equal:

Find the area of the parallelogram.

Explanation

Recall how to find the area of a parallelogram:

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

From the given parallelogram, we will need to use the Pythagorean Theorem to find the length of the height of the parallelogram.

$\text{Hypotenuse}^2=\text{Triangle base}^2+\text{height}^2$

$\text{height}^2=\text{Hypotenuse}^2-\text{Triangle base}^2$

$\text{height}=\sqrt{\text{Hypotenuse}^2-\text{Triangle base}^2}$

Plug in the given values to find the length of the height.

$\text{height}=\sqrt{5^2-2^2}=\sqrt{25-4}=\sqrt{21}$

Now, use the height to find the area of the parallelogram.

$\text{Area}=12\times\sqrt{21}=54.99$

Remember to round to places after the decimal.

An isosceles triangle has a perimeter of . If the base of the triangle is two less than two times the length of each leg, what is the height of the triangle?

The height of the triangle cannot be determined with the given information.

Explanation

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(6)-2=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-5^2}=\sqrt{11}=3.32$

Make sure to round to places after the decimal.

An acute isosceles triangle has an area of square units and a base with length . Find the perimeter of this triangle.

$2\sqrt{73} +6$

$\sqrt{73}+6$

$2\sqrt{73}+8$

$\sqrt{73}+8$

Explanation

To solve this problem, first work backwards using the formula:

$Area=\frac{b\cdot h}{2}$

Plugging in the given information and solving for height.

$24=\frac{6h}{2}$

$24\times2=6h$

$h=\frac{48}{6}=8$

Now that you've found the height of the triangle, use the Pythagorean Theorem to find the length of one of the equivalent sides of the triangle.

(Note, when applying the formula divide the measurement of the base in half--so the triangle will have sides of and .

Thus, the solution is:

$c=\sqrt{73}$

In this problem, the isosceles triangle will have two sides with the length of $\sqrt{73}$ and one side length of . Therefore, the perimeter is: $2\sqrt{73}+6$

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