Find the area of the shaded region.

Question

https://ibb.co/bjxZQDbF

Answer 1

Since the triangle contains the center of the circle and it contains a 30-degree angle, then that means it is also a 30-60-90 triangle.

In that 30-60-90 triangle, if the side opposite to 30 degrees is x, then the hypotenuse is 2x, and the side opposite of 60 degrees is 3 * sqrt x. Working backwards, then x = 9 (side opposite of 30 degrees), and the hypotenuse is 18 (also the diameter of the circle).

The area of the shaded region that you are trying to find is [half the area of the circle - area of the triangle]

= 1/2 * pi (9^2) - 1/2 * 9 * 9 sqrt 3

*See if you can solve for it a different way. Then you can truly own this problem. :)

Answer 2

I like this question a lot as it tests a few different key geometric identities and definitions.

The first thing to focus on is the orientation of the triangle inside the circle. One side is the diameter and one vertex touches the circumference of the circle, which reminds us of Thales Theorem that states the angle opposite the diameter MUST be 90 degrees!

Now that we have confirmed that the triangle is a right angled one, we calculate the third angle as 60 degrees (easy!). Then we notice another key clue; this triangle is the famous 30-60-90 triangle. This is a special triangle as its sides have the following relationship:

rule 1 - the side opposite the 30 degrees is the shortest

rule 2 - the side opposite 60 degrees is sqrt(3) times the shortest side

rule 3 - The side opposite the 90 degrees is the longest side and is twice the length of the shortest side.

Now we have all of the missing pieces to take this question to the finish. To do that, we will follow these steps in sequence:

1 - Calculate the diameter and then the radius of the circle.

2 - Calculate the area of the circle and divide it by half (as the triangle sits inside half of the circle area)

3 - Then calculate the area of the triangle.

4 - Finally, subtract the area of the triangle from the half circle area, which will give us the combined area as shaded in the figure.

Time to roll up the sleeves!

1 - The side opposite 60 degrees is 9 x sqrt(3) which must be sqrt(3) times the shortest side (see rule 2). This means the shortest side is (conveniently) 9

Diameter must be twice the shortest side (see rule 3). So the diameter D = 2 x 9 = 18. Then the radius (r) is D / 2 = 9

2 - Area of a circle is pi x r x r . Assuming pi = 3.14, area of the circle is A = (3.14)(9)(9) = 254.34. Area of the semi circle (a) is then a = A / 2 = 127.17. Nice!

3 - Area of the triangle (T) is simply half of the product of the two sides that are perpendicular to one another. So T = 0.5 x 9sqrt(3) x 9 = 70.15

4 - Final step is to subtract the area of the triangle from the area of the semi-circle. Before we do that we also notice the area of the triangle is less than that of the semi-circle, as it should be since the triangle sits INSIDE the semi-circle so it MUST be smaller. If it was larger, we would have to go back and check our calculation at this point before moving forward ;)

Since the numbers make sense, we can calculate the area of the shaded region (s) as: s = 127.17 - 70.15 = 57.02

Geometry is so much fun! :)

Answer 3

The easiest method for finding the area of the shaded region will be to subtract the area of the triangle from the area of the semicircle.

According to Thale's Theorem, any angle inscribed in a semicircle must be a right triangle with its right angle on the arc of the circle. (This is a specialized case of the Inscribed Angle Theorem.) Since we know that one angle of our triangle is 30°, we can use the Triangle Sum Theorem which states the the three interior angles of a triangle must add up to 180°. This means that the third angle of the triangle must measure 60° and that we must have a 30-60-90 triangle.

The diagram labels the longer leg of the triangle (opposite the 60° angle) as 9√3. According to the ratios of a 30-60-90 triangle, we can divide the long leg by √3 to get the length of the short leg (opposite the 30° angle). That gives us 9. The ratios also tell us that we can multiply the short leg of the triangle by 2 to get the length of the hypotenuse (opposite the 90° angle). That gives us 18.

So we know that the base of our triangle is 9, its height is 9√3, and the diameter of the circle is 18. Since the diameter is twice the length of the radius of a circle, our radius is 9.

Area of a triangle = 1/2bh = 1/2(9)(9√3) = 81√3/2

Area of semicircle = 1/2π r^2 = 1/2π(9)^2 = 81π/2

Shaded area = (semicircle area) - (triangle area)

= 81π/2 - 81√3/2

= 81/2 (π-√3)

≈ 57.0864

Answer 4

Our first objective should be to find the radius R of the circle.

From the figure, note that the triangle has one side as the diameter of the circle.

Let the radius of the circle be = R. This would make the longest side of the triangle is = 2R.

Based on the Inscribed Angle Theorem (An inscribed angle is half the measure of the central angle that subtends the same arc), the inscribed angle (which is the angle of the triangle at the point where it touches the circle) will be half of the angle made by the arc opposite to it.

The opposite arc is a semi-circle and makes a 180-degree angle at the center. Therefore, the inscribed angle is 90 degrees, which makes the inscribed triangle a right-angled triangle with hypotenuse = 2R.

We know from trigonometry that

Cos (30^o) = √3 / 2 --> (1)

From the diagram,

Cos (30^o) = 9 √3 / 2 R --> (2) - since Cosine of an angle in a right-angled triangle is equal to the adjacent side divided by the hypotenuse.

Putting (1) and (2) together,

√3 / 2 = 9 √3 / 2 R

This would give us R = 9

Using Sin(30^o) = 1/2, we can calculate the third side of the triangle to be = R = 9

Calculating the shaded area is really easy at this point.

The shaded area

= Area of the semi-circle minus the area of the triangle

= 1/2 * π * 9² - 1/2 * 9 * 9 √3

= 1/2 * 81 * (3.1416 - 1.7321)

= 40.5 * 1.4095

= 57.8868 square units

Note that with higher precision for π and √3, you will get a refined answer of 57.0864 square units.

Answer 5

In order to find the area of the shaded region, we would first need to find the area of the semicircle and the triangle, then subtract the area of the triangle from the area of the semicircle, and that would leave us with the area of the shaded region.

Step 1: find the area of the semicircle

Area of a circle is πr² so the semicircle is half that or (1/2)πr².

To find the radius, we have to solve for the hypotenuse of the triangle. This happens to be a right triangle because a triangle inscribed in a semicircle is always a right triangle. The right angle is touching the side of the circle, so the 3rd angle must be 60 degrees, since the angles inside of a triangle must add up to 180^ο (30+60+90=180). We can use the special right triangle formula for solving the hypotenuse (which happens to be the diameter of the circle). For a 30-60-90 triangle the formula for the respective sides are x, x√3, and 2x. The long leg is opposite the 60^ο angle, which is 9√3 = x√3. Therefore, x=9 (short leg), so the hypotenuse is 2x=2(9)=18, which is the length of the diameter. Half of the diameter is the radius, so the radius is equal to 9. The area of the semicircle is

(1/2)πr² = (1/2)(3.14)(9)² = 127.17

Step 2: Find the area of the triangle, which is (1/2)(base)(height). Since it's a right triangle, we can use the legs as the base and the height, since they are perpendicular to each other. We found these lengths in step 1, so we have (1/2)(9)(9√3) = 70.15

Step 3: Now we can subtract them to get the shaded region:

127.17 - 70.15 = 57.07 units squared

Answer 6

The triangle is a right triangle, and we know this because an angle inscribed in a semi-circle is a right angle. So this is a 30-60-90 triangle. We know the length of the longer leg, so we can use that to find the shorter leg, and to find the hypotenuse, which is also the diameter of the circle (because the point it goes through denotes the center).

The formula for a 30-60-90 triangle is s, s√3, 2s. Our longer leg is 9√3, so our shorter leg is 9 and our hypotenuse is 18.

For a right triangle, the two legs serve as the base and the height, which1 we need to find the area, A=1/2bh. For this triangle, that would be A=1/2(9)(9√3) or 40.5√3.

The semi-circle has an area formula is A=1/2(πr²). For this circle, that is A=1/2(π)(9²) or 40.5π.

To find the shaded area, we need to subtract these areas, so 40.5π-40.5√3 ≈ 57.086

Answer 7

2^x2-10(2^x)+16=0

first change to a quadratic equation first

let 2^x=y

y²-10y+16=0 now we have a quadratic equation

a_* c=product

a=1, b=1_* 16

sum=-10

now we find two numbers if multiplied together gives me 16 and if added together give -10

the numbers are -8,-2

continuation

y² -8y-2y+16=0

y(y-8)-2(y-8)

(y-2)(y-8)=0

y-2=0 y=2

y-8=0 y=8

now we let 2^x=y

y=2

y=8

when y=2

2^x=2¹

^x=1

^y=8

2^x=8

2^x=2³

x=3

Note that in indices if the bases are the same the powers are also similar

Answer 8