1) Find the possible values of a, if the distance between the points is 5, and the coordinates are (1, 1) and (4, a). I don't know how to solve this one.
2) in a right triangle the length of the hypotenuse is 14 and the length b of one of the legs is 8. find the length a of the other leg to the nearest hundredth.
3) Which of the following side measures would form a right triangle?
A. 7, 9 ,12
B. 10, 24, 26
C. 18, 20, 27
D. 8, 11, 19
I said A
4) Determine the third side measure that will form a right triangle if two sides are 7 and 24.
I said 25
5) Find the missing side of the right triangle with sides: h = 5, a = 3
I said 5.8
6) Can two sets of data have the same mean, but not the same variance.
I said yes
7) A group of 15 of your friends wants to go to a concert. Unfortunately, you only have 10 tickets available to give to your friends. How many different groups of friends could you take to the concert?
I'm not to good with probability, so I'm not too sure about this one
1.
Use the distance formula:
distance =
(4-1)²+(a-1)²=5²
Solve for a (a quadratic equation)
2.
Use Pythagorean formula:
8²+b²=14²
Solve for b.
3(a)
7²+9²=49+81 = 130 ≠ 12²
Therefore the three sides do not form a right triangle.
Try the others similarly.
Post your answer for checks if you wish.
Consider the sequence of steps to solve the equation:
x - 3
3
= x + 3
Step 1 ⇒ x - 3 = 3(x + 3)
Step 2 ⇒ x - 3 = 3x + 9
Step 3 ⇒ x = 3x + 12
Step 4 ⇒ -2x = 12
Step 5 ⇒ x = -6
Identify the property of equality which yields Step 5.
1) To find the possible values of a in the given scenario, we need to use the distance formula. The distance formula is calculated as follows:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of the two points are (1, 1) and (4, a). So, using the distance formula, we can set up the following equation:
5 = sqrt((4 - 1)^2 + (a - 1)^2)
Now, let's solve for a. First, we square both sides of the equation to eliminate the square root:
25 = (4 - 1)^2 + (a - 1)^2
Simplifying the right side:
25 = 9 + (a - 1)^2
Subtracting 9 from both sides:
16 = (a - 1)^2
Taking the square root of both sides:
±4 = a - 1
Now, we can solve for a by considering both the positive and negative square roots separately:
For the positive square root:
4 = a - 1
a = 5
For the negative square root:
-4 = a - 1
a = -3
Therefore, the possible values of a are 5 and -3.
2) In a right triangle, the Pythagorean theorem can be used to find the lengths of the sides. The Pythagorean theorem is given by:
a^2 + b^2 = c^2
In this case, the length of the hypotenuse (c) is given as 14, and the length of one of the legs (b) is given as 8. Let's substitute these values into the equation:
a^2 + 8^2 = 14^2
Simplifying the equation:
a^2 + 64 = 196
Subtracting 64 from both sides:
a^2 = 132
Taking the square root of both sides:
a ≈ ±11.49
Since we are looking for the length of a side, we disregard the negative value. Therefore, the length a is approximately 11.49 (rounded to the nearest hundredth).
3) To determine which set of side measures would form a right triangle, we need to check if the Pythagorean theorem holds true.
For option A (7, 9, 12):
7^2 + 9^2 = 49 + 81 = 130
12^2 = 144
Since 130 is not equal to 144, option A does not form a right triangle.
For option B (10, 24, 26):
10^2 + 24^2 = 100 + 576 = 676
26^2 = 676
Since both sides of the equation are equal, option B forms a right triangle.
For option C (18, 20, 27):
18^2 + 20^2 = 324 + 400 = 724
27^2 = 729
Since 724 is not equal to 729, option C does not form a right triangle.
For option D (8, 11, 19):
8^2 + 11^2 = 64 + 121 = 185
19^2 = 361
Since 185 is not equal to 361, option D does not form a right triangle.
Therefore, the correct option is B, with side measures 10, 24, and 26, which form a right triangle.
4) To determine the third side measure that will form a right triangle if two sides are 7 and 24, we can use the Pythagorean theorem.
Let's assume the third side measure is represented by c. The Pythagorean theorem can be written as:
7^2 + 24^2 = c^2
49 + 576 = c^2
625 = c^2
Taking the square root of both sides:
c ≈ ±25
Since we cannot have a negative length for a side, the third side measure is 25.
5) To find the missing side of a right triangle with known lengths of h = 5 and a = 3, we can use the Pythagorean theorem.
The Pythagorean theorem can be written as:
a^2 + b^2 = c^2
In this case, we know a = 3 and c = 5. Let's substitute these values into the equation:
3^2 + b^2 = 5^2
9 + b^2 = 25
Subtracting 9 from both sides:
b^2 = 16
Taking the square root of both sides:
b ≈ ±4
Since we are dealing with side lengths, we disregard the negative value. Therefore, the missing side length (b) is approximately 4.
6) Yes, two sets of data can have the same mean but not the same variance. The mean is a measure of central tendency and represents the average value of a set of data. Variance, on the other hand, measures the spread or dispersion of the data points around the mean.
Consider two sets of data:
Set A: {1, 2, 3, 4, 5}
Set B: {1, 2, 4, 4, 5}
Both sets have the same mean, which is 3, but their variances are different. The variance of set A is 2, while the variance of set B is 1.6. This illustrates that even though the mean is the same, the distribution of the data points can vary, leading to different variances.
7) To determine how many different groups of friends could attend the concert when there are only 10 tickets available for 15 friends, we can use combinations.
The number of different groups can be found using the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where n represents the total number of friends (15) and r represents the number of tickets available (10).
Plugging in the values:
C(15, 10) = 15! / (10! * (15-10)!)
C(15, 10) = (15 * 14 * 13 * 12 * 11) / (10 * 9 * 8 * 7 * 6)
C(15, 10) = 3,003
Therefore, there are 3,003 different groups of friends that could attend the concert.