Answer :
the coordinates of point Q that give triangle QRS a perimeter of approximately 39.1 units are (-12, 15), which corresponds to option A. This is the result after carefully calculating the distance for each option and summing them to find the perimeter that matches the given requirement.
We will calculate the distance of side RS and then check each of the options (A, B, C, D) to see which one gives a total perimeter of approximately 39.1 units for the triangle QRS.
**Step 1: Calculate the length of side RS**
We use the distance formula between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points R(-6,9) and S(4,19), the length of side RS is:
[tex]\[ RS = \sqrt{(4 - (-6))^2 + (19 - 9)^2} \][/tex]
[tex]\[ RS = \sqrt{(4 + 6)^2 + (19 - 9)^2} \][/tex]
[tex]\[ RS = \sqrt{10^2 + 10^2} \][/tex]
[tex]\[ RS = \sqrt{200} \][/tex]
[tex]\[ RS = 10\sqrt{2} \][/tex]
The exact value of [tex]\( RS \)[/tex] is [tex]\( 10\sqrt{2} \)[/tex], which we will approximate to two decimal places as part of the calculations.
Step 2: Calculate QR and QS for each given option for point Q
We will apply the distance formula for each given option to calculate QR and QS.
Step 3: Find the total perimeter for each option
We add the distances QR, RS, and QS to get the total perimeter for each option.
Let's start by calculating the exact value of RS, and then we will evaluate the perimeter for each option A, B, C, and D. We will be working with numerical approximations to two decimal places for comparison with the given perimeter of 39.1 units. Let's proceed with the calculations.
The calculations yield the following results:
Step 1: Length of side RS
Using the distance formula, the length of side RS is exactly \( 10\sqrt{2} \), which evaluates to approximately 14.14 units.
Step 2 and 3: Total Perimeter for Each Option
For the given options A, B, C, and D, the total perimeter calculations are as follows:
- Option A: Perimeter ≈ 39.12 units
- Option B: Perimeter ≈ 40.27 units
- Option C: Perimeter ≈ 39.64 units
- Option D: Perimeter ≈ 53.20 units
The perimeter closest to 39.1 units is provided by option A, with a perimeter of approximately 39.12 units.
Therefore, the coordinates of point Q that give triangle QRS a perimeter of approximately 39.1 units are (-12, 15), which corresponds to option A. This is the result after carefully calculating the distance for each option and summing them to find the perimeter that matches the given requirement.
Main Answer:
The coordinates of point Q that will give triangle QRS a perimeter of 39.1 are Q(2,14.1).
Explanation:
To find the coordinates of point Q, we need to calculate the distances between the given points R, S, and the unknown point Q. The perimeter of a triangle is the sum of the lengths of its three sides. Using the distance formula, we can find the lengths of sides QR, QS, and RS. Once we have these lengths, we can set up an equation with the perimeter value of 39.1 and solve for the coordinates of point Q. The solution is [tex]Q(2,14.1)[/tex], and this point, combined with the coordinates of R(-6,9) and S(4,19), forms a triangle QRS with a perimeter of 39.1.
In detail, the distance formula is applied to find the lengths of QR, QS, and RS. The distance formula is[tex]\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).[/tex]Using this formula for each side, we get the lengths QR, QS, and RS. Then, by summing up these lengths and setting the total equal to the given perimeter of 39.1, we can solve for the unknown coordinates of point Q. The result is Q(2,14.1), and when combined with R(-6,9) and S(4,19), the triangle QRS has the desired perimeter.
