Question 44 - Page 11
Two Lines in the Plane
Progress: 0 out of 4 parts
Given two lines:
y = 3x + 9 and y = -3x + 9
A - the intersection point of the two lines
B - where the line y=3x+9 meets the x-axis
C - where the line y=-3x+9 meets the x-axis
y = 3x + 9 and y = -3x + 9
A - the intersection point of the two lines
B - where the line y=3x+9 meets the x-axis
C - where the line y=-3x+9 meets the x-axis
a. Find the coordinates of points A, B, C
a
What are the coordinates of the points?
A(
,
)
B(
,
)
C(
,
)
Excellent! You found all the points!
A(0,9) - Intersection: set 3x+9 = -3x+9, which gives x=0, y=9
B(-3,0) - x-axis intercept: set y=0 in 3x+9=0, giving x=-3
C(3,0) - x-axis intercept: set y=0 in -3x+9=0, giving x=3
B(-3,0) - x-axis intercept: set y=0 in 3x+9=0, giving x=-3
C(3,0) - x-axis intercept: set y=0 in -3x+9=0, giving x=3
b. Find the lengths of segments AO, BO, CO
b
What are the segment lengths? (O is the origin)
AO =
BO =
CO =
Very correct!
AO = 9 (distance along the y-axis from (0,0) to (0,9))
BO = 3 (distance along the x-axis from (-3,0) to (0,0))
CO = 3 (distance along the x-axis from (0,0) to (3,0))
BO = 3 (distance along the x-axis from (-3,0) to (0,0))
CO = 3 (distance along the x-axis from (0,0) to (3,0))
c. Prove: ΔABO ≅ ΔACO
c
Choose the appropriate congruence theorem:
Great! SAS is the correct answer!
Proof of congruence:
• AO is shared by both triangles
• OB = OC = 3
• ∠AOB = ∠AOC = 90° (AO is on the y-axis, and OB/OC are on the x-axis)
Therefore congruent by SAS → ΔABO ≅ ΔACO
• AO is shared by both triangles
• OB = OC = 3
• ∠AOB = ∠AOC = 90° (AO is on the y-axis, and OB/OC are on the x-axis)
Therefore congruent by SAS → ΔABO ≅ ΔACO
d. Calculate the area of ΔABC
d
What is the area of triangle ABC?
Base BC =
Height =
Area =
Excellent! You calculated the area correctly!
The base BC lies on the x-axis: BC = 3 - (-3) = 6
The height from A to the x-axis is 9
Therefore: Area = (1/2) × 6 × 9 = 27
The height from A to the x-axis is 9
Therefore: Area = (1/2) × 6 × 9 = 27
Well done! You finished the question!
You understood the connection between linear functions, intersection points, and triangle congruence!