the vertices of angle PQR are P(3,0),Q(9,-2) and R(9,8)find the area of angle PQR to one decimal place and state the type of triangle
2/8 of 15
sketch it first
length QR = 10 NOTE - perpendicular to x axis so altitude along x axis
altitude from P to QR is along x axis and of length 9-3 = 6
so
area = (1/2)(6)(10) = 30
Huh Mauricio ????
To find the area of triangle PQR, we can use the formula for the area of a triangle given the coordinates of its vertices.
Step 1: Find the length of two sides of the triangle.
- PQ: Distance between points P(3,0) and Q(9,-2). We use the distance formula: d(PQ) = √((x2 - x1)^2 + (y2 - y1)^2)
d(PQ) = √((9 - 3)^2 + (-2 - 0)^2)
d(PQ) = √(6^2 + (-2)^2)
d(PQ) = √(36 + 4)
d(PQ) = √40 = 2√10 (approx.)
- PR: Distance between points P(3,0) and R(9,8).
d(PR) = √((9 - 3)^2 + (8 - 0)^2)
d(PR) = √(6^2 + 8^2)
d(PR) = √(36 + 64)
d(PR) = √100 = 10
Step 2: Calculate the area of triangle PQR using the lengths of PQ and PR.
- Area of triangle PQR = (1/2) * PQ * PR
Area = (1/2) * (2√10) * (10)
Area = √10 * 10
Area = 10√10 (approx.)
Therefore, the area of triangle PQR is approximately 10√10 square units. To one decimal place, it is approximately 31.6 square units.
Now, let's determine the type of triangle PQR. We can do this by examining the lengths of its sides:
- Since PQ = 2√10 (approx.), PR = 10, and QR = √[(9 - 9)^2 + (8 - (-2))^2] = √100 = 10, we can determine the type of triangle based on its side lengths.
- If all sides have different lengths, it is a scalene triangle.
- If two sides have the same length, it is an isosceles triangle.
- If all sides have the same length, it is an equilateral triangle.
In this case, since all three sides of triangle PQR have different lengths (PQ ≠ PR ≠ QR), it is a scalene triangle.