Question 2

Find and record the [tex]\(x\)[/tex]-coordinate of a point that divides the horizontal side in the ratio [tex]\(2:3\)[/tex]. Use the formula from the lesson to guide you:

[tex]\[
x_{C} = \frac{a x_{A} + b x_{B}}{a + b}
\][/tex]



Answer :

To solve for the [tex]\( x \)[/tex]-coordinate of a point [tex]\( C \)[/tex] that divides the horizontal side in the ratio [tex]\( 2:3 \)[/tex], we will use the given formula:
[tex]\[ x_C = \frac{a x_a + b z_A}{a + b} \][/tex]

Here, the following values are provided:
- [tex]\( a = 2 \)[/tex] (the first part of the ratio)
- [tex]\( b = 3 \)[/tex] (the second part of the ratio)
- [tex]\( x_a = 1 \)[/tex] (the [tex]\( x \)[/tex]-coordinate of point [tex]\( A \)[/tex])
- [tex]\( z_A = 4 \)[/tex] (the [tex]\( z \)[/tex]-coordinate of point [tex]\( B \)[/tex])

Now we substitute these values into the formula:

[tex]\[ x_C = \frac{(2 \times 1) + (3 \times 4)}{2 + 3} \][/tex]

Let's break this down step by step:

1. Calculate the product of [tex]\( a \)[/tex] and [tex]\( x_a \)[/tex]:
[tex]\[ 2 \times 1 = 2 \][/tex]

2. Calculate the product of [tex]\( b \)[/tex] and [tex]\( z_A \)[/tex]:
[tex]\[ 3 \times 4 = 12 \][/tex]

3. Add the results of these products:
[tex]\[ 2 + 12 = 14 \][/tex]

4. Sum the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ 2 + 3 = 5 \][/tex]

5. Finally, divide the sum of the products by the sum of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x_C = \frac{14}{5} = 2.8 \][/tex]

Therefore, the [tex]\( x \)[/tex]-coordinate of the point [tex]\( C \)[/tex] that divides the horizontal side in the ratio [tex]\( 2:3 \)[/tex] is [tex]\( 2.8 \)[/tex].