Answer :
Sure, let's address each question one by one.
### 7) If [tex]\( A \widehat{B C} = A \widehat{C B} \)[/tex] and [tex]\( A \widehat{B D} = A \widehat{C D} \)[/tex], show that [tex]\( D \widehat{B C} = D \widehat{C B} \)[/tex].
Given:
1. [tex]\( A \widehat{B C} = A \widehat{C B} \)[/tex]
2. [tex]\( A \widehat{B D} = A \widehat{C D} \)[/tex]
We need to show that [tex]\( D \widehat{B C} = D \widehat{C B} \)[/tex].
From the given information:
- [tex]\(A \widehat{B C} = A \widehat{C B}\)[/tex] implies that the angle formed at point [tex]\(A\)[/tex] between points [tex]\(B\)[/tex] and [tex]\(C\)[/tex] is the same regardless of the order.
- [tex]\(A \widehat{B D} = A \widehat{C D}\)[/tex] implies that the angle formed at point [tex]\(A\)[/tex] between points [tex]\(B(D)\)[/tex] and [tex]\(D(C)\)[/tex] is also the same.
Since [tex]\(D\)[/tex] is common in both angles and the angles formed with [tex]\(B\)[/tex] and [tex]\(C\)[/tex] are equal at point [tex]\(A\)[/tex], it implies symmetrical properties for the angles involving [tex]\(D\)[/tex] and both [tex]\(B\)[/tex] and [tex]\(C\)[/tex]. Thus, it can be inferred that the angle formed at [tex]\(D\)[/tex] between [tex]\(B\)[/tex] and [tex]\(C\)[/tex] must be equal to the angle formed at [tex]\(D\)[/tex] between [tex]\(C\)[/tex] and [tex]\(B\)[/tex].
Therefore, [tex]\(D \widehat{B C} = D \widehat{C B}\)[/tex].
### 8) Solve the equation
[tex]\[ \frac{2(3 x - 1)}{5} - 3 = 5 \][/tex]
First, we want to simplify and isolate [tex]\(x\)[/tex].
1. Start by eliminating the constant on the right side:
[tex]\[ \frac{2(3 x - 1)}{5} - 3 = 5 \][/tex]
Add [tex]\(3\)[/tex] to both sides:
[tex]\[ \frac{2(3 x - 1)}{5} = 8 \][/tex]
2. Next, eliminate the fraction by multiplying both sides by [tex]\(5\)[/tex]:
[tex]\[ 2(3 x - 1) = 40 \][/tex]
3. Distribute the [tex]\(2\)[/tex] on the left side:
[tex]\[ 6x - 2 = 40 \][/tex]
4. Move the constant term to the other side by adding [tex]\(2\)[/tex]:
[tex]\[ 6x = 42 \][/tex]
5. Finally, solve for [tex]\(x\)[/tex] by dividing by [tex]\(6\)[/tex]:
[tex]\[ x = 7 \][/tex]
### 9) In the formula [tex]\( v = u + at \)[/tex], show [tex]\( t \)[/tex] as the subject.
We start with the equation:
[tex]\[ v = u + at \][/tex]
To make [tex]\(t\)[/tex] the subject, isolate [tex]\(t\)[/tex].
1. Subtract [tex]\(u\)[/tex] from both sides:
[tex]\[ v - u = at \][/tex]
2. Divide both sides by [tex]\(a\)[/tex]:
[tex]\[ t = \frac{v - u}{a} \][/tex]
Thus, the subject [tex]\(t\)[/tex] is:
[tex]\[ t = \frac{v - u}{a} \][/tex]
### 10) Find the value of [tex]\(x\)[/tex].
No specific equation or context is given for [tex]\(x\)[/tex], so it can't be solved without more information.
### 11) If [tex]\( x = -3 \)[/tex] and [tex]\( y = -\frac{1}{2} \)[/tex], find the value of [tex]\( x^2 - 8y \)[/tex].
Given:
[tex]\( x = -3 \)[/tex]
[tex]\( y = -\frac{1}{2} \)[/tex]
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = (-3)^2 = 9 \][/tex]
2. Calculate [tex]\( 8y \)[/tex]:
[tex]\[ 8y = 8 \cdot \left(-\frac{1}{2}\right) = -4 \][/tex]
3. Find the value of [tex]\( x^2 - 8y \)[/tex]:
[tex]\[ x^2 - 8y = 9 - (-4) = 9 + 4 = 13 \][/tex]
So, the value of [tex]\( x^2 - 8y \)[/tex] is [tex]\( 13 \)[/tex].
### 7) If [tex]\( A \widehat{B C} = A \widehat{C B} \)[/tex] and [tex]\( A \widehat{B D} = A \widehat{C D} \)[/tex], show that [tex]\( D \widehat{B C} = D \widehat{C B} \)[/tex].
Given:
1. [tex]\( A \widehat{B C} = A \widehat{C B} \)[/tex]
2. [tex]\( A \widehat{B D} = A \widehat{C D} \)[/tex]
We need to show that [tex]\( D \widehat{B C} = D \widehat{C B} \)[/tex].
From the given information:
- [tex]\(A \widehat{B C} = A \widehat{C B}\)[/tex] implies that the angle formed at point [tex]\(A\)[/tex] between points [tex]\(B\)[/tex] and [tex]\(C\)[/tex] is the same regardless of the order.
- [tex]\(A \widehat{B D} = A \widehat{C D}\)[/tex] implies that the angle formed at point [tex]\(A\)[/tex] between points [tex]\(B(D)\)[/tex] and [tex]\(D(C)\)[/tex] is also the same.
Since [tex]\(D\)[/tex] is common in both angles and the angles formed with [tex]\(B\)[/tex] and [tex]\(C\)[/tex] are equal at point [tex]\(A\)[/tex], it implies symmetrical properties for the angles involving [tex]\(D\)[/tex] and both [tex]\(B\)[/tex] and [tex]\(C\)[/tex]. Thus, it can be inferred that the angle formed at [tex]\(D\)[/tex] between [tex]\(B\)[/tex] and [tex]\(C\)[/tex] must be equal to the angle formed at [tex]\(D\)[/tex] between [tex]\(C\)[/tex] and [tex]\(B\)[/tex].
Therefore, [tex]\(D \widehat{B C} = D \widehat{C B}\)[/tex].
### 8) Solve the equation
[tex]\[ \frac{2(3 x - 1)}{5} - 3 = 5 \][/tex]
First, we want to simplify and isolate [tex]\(x\)[/tex].
1. Start by eliminating the constant on the right side:
[tex]\[ \frac{2(3 x - 1)}{5} - 3 = 5 \][/tex]
Add [tex]\(3\)[/tex] to both sides:
[tex]\[ \frac{2(3 x - 1)}{5} = 8 \][/tex]
2. Next, eliminate the fraction by multiplying both sides by [tex]\(5\)[/tex]:
[tex]\[ 2(3 x - 1) = 40 \][/tex]
3. Distribute the [tex]\(2\)[/tex] on the left side:
[tex]\[ 6x - 2 = 40 \][/tex]
4. Move the constant term to the other side by adding [tex]\(2\)[/tex]:
[tex]\[ 6x = 42 \][/tex]
5. Finally, solve for [tex]\(x\)[/tex] by dividing by [tex]\(6\)[/tex]:
[tex]\[ x = 7 \][/tex]
### 9) In the formula [tex]\( v = u + at \)[/tex], show [tex]\( t \)[/tex] as the subject.
We start with the equation:
[tex]\[ v = u + at \][/tex]
To make [tex]\(t\)[/tex] the subject, isolate [tex]\(t\)[/tex].
1. Subtract [tex]\(u\)[/tex] from both sides:
[tex]\[ v - u = at \][/tex]
2. Divide both sides by [tex]\(a\)[/tex]:
[tex]\[ t = \frac{v - u}{a} \][/tex]
Thus, the subject [tex]\(t\)[/tex] is:
[tex]\[ t = \frac{v - u}{a} \][/tex]
### 10) Find the value of [tex]\(x\)[/tex].
No specific equation or context is given for [tex]\(x\)[/tex], so it can't be solved without more information.
### 11) If [tex]\( x = -3 \)[/tex] and [tex]\( y = -\frac{1}{2} \)[/tex], find the value of [tex]\( x^2 - 8y \)[/tex].
Given:
[tex]\( x = -3 \)[/tex]
[tex]\( y = -\frac{1}{2} \)[/tex]
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = (-3)^2 = 9 \][/tex]
2. Calculate [tex]\( 8y \)[/tex]:
[tex]\[ 8y = 8 \cdot \left(-\frac{1}{2}\right) = -4 \][/tex]
3. Find the value of [tex]\( x^2 - 8y \)[/tex]:
[tex]\[ x^2 - 8y = 9 - (-4) = 9 + 4 = 13 \][/tex]
So, the value of [tex]\( x^2 - 8y \)[/tex] is [tex]\( 13 \)[/tex].
