In the diagram, ABC is an equilateral triangle. Find the value of (x+y).

Question

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Here are the solutions to the problems:

:

a) Given that $P = \{\text{factors of } 36\}$ and $Q = \{\text{factors of } 54\}$ .

i) List the members in the sets $P$ and $Q$ .

List the factors of 36.

$$
P = \{1, 2, 3, 4, 6, 9, 12, 18, 36\}
$$

List the factors of 54.

$$
Q = \{1, 2, 3, 6, 9, 18, 27, 54\}
$$
The members are:  
$P = \boxed{\{1, 2, 3, 4, 6, 9, 12, 18, 36\}}$  
$Q = \boxed{\{1, 2, 3, 6, 9, 18, 27, 54\}}$

ii) Find:
- ( $\alpha$ ) $P \cap Q$ ;

Identify the common factors in sets $P$ and $Q$ .

    $$
    P \cap Q = \{1, 2, 3, 6, 9, 18\}
    $$
    $P \cap Q = \boxed{\{1, 2, 3, 6, 9, 18\}}$

*   **($\beta$) $n(P \cap Q)$;**

Count the number of elements in $P \cap Q$ .

    There are 6 elements in $P \cap Q$.  
    $n(P \cap Q) = \boxed{6}$

*   **($\gamma$) the Highest Common Factor (HCF) of 36 and 54.**

The HCF is the largest number in the set $P \cap Q$ .
The largest common factor is 18. HCF of 36 and 54 is $\boxed{18}$.
b) Write down the next two terms of the sequence 1, 4, 9, ...
Identify the pattern of the sequence.
The sequence consists of perfect squares: $1^2=1$ , $2^2=4$ , $3^2=9$ .
Calculate the next two terms.
The next terms will be $4^2$ and $5^2$ .
$4^2 = 16$
$5^2 = 25$
The next two terms are $\boxed{16, 25}$ .

c) The median of the ordered set of observations 2, 3, $(4m - 3)$ , $(3m + 1)$ , 11 and 13 in ascending order is 6. Find the value of $m$ .
Identify the middle terms.
There are 6 observations, so the median is the average of the 3rd and 4th terms.
The 3rd term is $(4m - 3)$ and the 4th term is $(3m + 1)$ .
Set up the equation for the median.
$\text{Median} = \frac{(4m - 3) + (3m + 1)}{2}$
Given that the median is 6:
$6 = \frac{4m - 3 + 3m + 1}{2}$
Solve for $m$ .
$6 = \frac{7m - 2}{2}$ $12 = 7m - 2$ $12 + 2 = 7m$ $14 = 7m$ $m = \frac{14}{7}$ $m = 2$
The value of $m$ is $\boxed{2}$ .

:

a) Simplify: $\left(\frac{1}{3} + \frac{1}{12}\right) \div \left(\frac{2}{3} - \frac{5}{8}\right)$ .
Simplify the first parenthesis.
$\frac{1}{3} + \frac{1}{12} = \frac{4}{12} + \frac{1}{12} = \frac{5}{12}$
Simplify the second parenthesis.
The least common multiple of 3 and 8 is 24.
$\frac{2}{3} - \frac{5}{8} = \frac{2 \times 8}{24} - \frac{5 \times 3}{24} = \frac{16}{24} - \frac{15}{24} = \frac{1}{24}$
Perform the division.
$\frac{5}{12} \div \frac{1}{24} = \frac{5}{12} \times \frac{24}{1}$ $= 5 \times 2 = 10$
The simplified expression is $\boxed{10}$ .

b) Find the product of $(2x - 3)$ and $(2x + 3)$ .
Recognize the pattern as a difference of squares, $(a-b)(a+b) = a^2 - b^2$ .
Here, $a = 2x$ and $b = 3$ .
Apply the formula.
$(2x - 3)(2x + 3) = (2x)^2 - 3^2$ $= 4x^2 - 9$
The product is $\boxed{4x^2 - 9}$ .

c) In the diagram, $ABC$ is an equilateral triangle. Find the value of $(x+y)$ .
The sides of the triangle are $AB = (3x+2)$ cm, $AC = 11$ cm, and $BC = (2y+3)$ cm.
Use the property of an equilateral triangle that all sides are equal.
$3x + 2 = 11$ $2y + 3 = 11$
Solve the equation for $x$ .
$3x + 2 = 11$ $3x = 11 - 2$ $3x = 9$ $x = \frac{9}{3}$ $x = 3$
Solve the equation for $y$ .
$2y + 3 = 11$ $2y = 11 - 3$ $2y = 8$ $y = \frac{8}{2}$ $y = 4$
Calculate $(x+y)$ .
$x+y = 3+4 = 7$
The value of $(x+y)$ is $\boxed{7}$ .

:

a) Given the relation $L = \frac{2(m^2 - n^2)}{4(m + n)}$ .
- i) Simplify $L$ .
Step 20

In the diagram, ABC is an equilateral triangle. Find the value of (x+y).

Step-by-step solution

List the factors of 36.

List the factors of 54.

Identify the common factors in sets $P$ and $Q$ .

Count the number of elements in $P \cap Q$ .

The HCF is the largest number in the set $P \cap Q$ .

Identify the pattern of the sequence.

Calculate the next two terms.

Identify the middle terms.

Set up the equation for the median.

Solve for $m$ .

Simplify the first parenthesis.

Simplify the second parenthesis.

Perform the division.

Recognize the pattern as a difference of squares, $(a-b)(a+b) = a^2 - b^2$ .

Apply the formula.

Use the property of an equilateral triangle that all sides are equal.

Solve the equation for $x$ .

Solve the equation for $y$ .

Calculate $(x+y)$ .

Step 20

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More Mathematics Questions

In the diagram, ABC is an equilateral triangle. Find the value of (x+y).

Step-by-step solution

List the factors of 36.

List the factors of 54.

Identify the common factors in sets PPP and QQQ.

Count the number of elements in P∩QP \cap QP∩Q.

The HCF is the largest number in the set P∩QP \cap QP∩Q.

Identify the pattern of the sequence.

Calculate the next two terms.

Identify the middle terms.

Set up the equation for the median.

Solve for mmm.

Simplify the first parenthesis.

Simplify the second parenthesis.

Perform the division.

Recognize the pattern as a difference of squares, (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2.

Apply the formula.

Use the property of an equilateral triangle that all sides are equal.

Solve the equation for xxx.

Solve the equation for yyy.

Calculate (x+y)(x+y)(x+y).

Step 20

Still stuck on this one?

More Mathematics Questions

Identify the common factors in sets $P$ and $Q$ .

Count the number of elements in $P \cap Q$ .

The HCF is the largest number in the set $P \cap Q$ .

Solve for $m$ .

Recognize the pattern as a difference of squares, $(a-b)(a+b) = a^2 - b^2$ .

Solve the equation for $x$ .

Solve the equation for $y$ .

Calculate $(x+y)$ .