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Mathematics Standard 2 1 - HSC - SSCE Mathematics Standard - Question 1 - 2023 - Paper 1

Question 1

Mathematics Standard 2 1. Use the following figure to answer the questions below. ![Figure](URL-to-the-figure) a) Calculate the area of the triangle formed ... show full transcript

Worked Solution & Example Answer:Mathematics Standard 2 1 - HSC - SSCE Mathematics Standard - Question 1 - 2023 - Paper 1

Step 1

a) Calculate the area of the triangle formed by the vertices A(2,3), B(5,11), and C(12,5).

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Answer

To calculate the area of a triangle given its vertices, we can use the formula:

Area = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|

Substituting the coordinates of points A(2,3), B(5,11), and C(12,5):

$x_1 = 2$ , $y_1 = 3$
$x_2 = 5$ , $y_2 = 11$
$x_3 = 12$ , $y_3 = 5$

Calculating:

Area = \frac{1}{2} \left| 2(11 - 5) + 5(5 - 3) + 12(3 - 11) \right| \ = \frac{1}{2} \left| 2(6) + 5(2) + 12(-8) \right| \ = \frac{1}{2} \left| 12 + 10 - 96 \right| \ = \frac{1}{2} \left| -74 \right| \ = 37

Thus, the area of the triangle is 37 square units.

Step 2

b) Determine the length of side AB.

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Answer

To determine the length of side AB, we use the distance formula:

Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Substituting the coordinates of points A(2,3) and B(5,11):

Distance_{AB} = \sqrt{(5 - 2)^2 + (11 - 3)^2} \ = \sqrt{3^2 + 8^2} \ = \sqrt{9 + 64} \ = \sqrt{73} \ \approx 8.54

Thus, the length of side AB is approximately 8.54 units.

Step 3

c) If the triangle is reflected across the y-axis, what are the coordinates of the new vertices?

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Answer

To find the coordinates of the reflected points across the y-axis, we change the sign of the x-coordinates.

The vertices are:

A(2,3) → A'(-2,3)
B(5,11) → B'(-5,11)
C(12,5) → C'(-12,5)

Thus, the new coordinates after reflection are A'(-2,3), B'(-5,11), and C'(-12,5).

Mathematics Standard 2 1 - HSC - SSCE Mathematics Standard - Question 1 - 2023 - Paper 1

Worked Solution & Example Answer:Mathematics Standard 2 1 - HSC - SSCE Mathematics Standard - Question 1 - 2023 - Paper 1

a) Calculate the area of the triangle formed by the vertices A(2,3), B(5,11), and C(12,5).

b) Determine the length of side AB.

c) If the triangle is reflected across the y-axis, what are the coordinates of the new vertices?

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