In △PQR, PQ = 17 in, PR = 10 in, QR = 21 in. Find the altitude from P to the side QR.
use the law of cosines to get, say angle R.
17^2 = 10^2 + 21^2 - 2*10*21 cosR
cosR = 3/5
Now find the altitude h from P using
h/10 = sinR = 4/5
h = 8
To find the altitude from P to the side QR in △PQR, we can use the formula for the area of a triangle.
Step 1: Calculate the area of the triangle.
The area of a triangle can be found using the formula:
Area = (base * height) / 2
In this case, the base is QR, and we need to find the height (altitude) from P to QR.
Step 2: Find the semi-perimeter of the triangle.
The semi-perimeter of a triangle can be calculated by adding the lengths of all three sides and dividing by 2.
Semi-perimeter = (PQ + PR + QR) / 2
Step 3: Use Heron's formula to find the area of the triangle.
Heron's formula states that the area of a triangle can be calculated using the semi-perimeter and the lengths of its sides.
Area = √(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter, and a, b, c are the lengths of the sides.
Step 4: Substitute the values into the formula.
In this case, PQ = 17, PR = 10, QR = 21.
Semi-perimeter = (17 + 10 + 21) / 2
= 48 / 2
= 24 in
Area = √(24 * (24 - 17) * (24 - 10) * (24 - 21))
= √(24 * 7 * 14 * 3)
= √(7056)
= 84 in²
Step 5: Calculate the altitude from P to QR.
The area of a triangle can also be calculated using the formula:
Area = (base * height) / 2
Rearranging the formula, we can find the height (altitude):
Height = (2 * Area) / base
In this case, the base is QR, and the area is 84 in².
Height = (2 * 84) / 21
= 168 / 21
= 8 in
Therefore, the altitude from P to the side QR is 8 inches.
To find the altitude from point P to side QR in triangle PQR, we can use the formula for the area of a triangle.
The area of a triangle is given by the formula:
Area = (base * height) / 2
Here, the side QR is the base of the triangle, and we need to find the height (which is the altitude from P to side QR).
Step 1: Calculate the semi-perimeter
The semi-perimeter of a triangle is given by the formula:
s = (a + b + c) / 2
In this case, a = PQ = 17 in, b = PR = 10 in, c = QR = 21 in.
So, the semi-perimeter s = (17 + 10 + 21) / 2 = 48 / 2 = 24 in.
Step 2: Apply Heron's formula to find the area
Heron's formula states that the area of a triangle can be calculated using the semi-perimeter and the lengths of the sides.
The area of the triangle PQR is given by:
Area = √(s(s - a)(s - b)(s - c))
So, plugging in the values, we have:
Area = √(24(24 - 17)(24 - 10)(24 - 21))
Calculating this, we get:
Area = √(24 * 7 * 14 * 3) = √(2^3 * 2 * 7 * 7 * 2 * 3) = √(2^4 * 7^2 * 3) = 2 * 7 * √6 = 14√6
Step 3: Calculate the altitude
To find the altitude, we need to divide the area by the base length.
Altitude = Area / base length
= 14√6 / 21
= (14 / 21)√6
= (2 / 3)√6
So, the altitude from point P to side QR in triangle PQR is (2/3)√6 inches.
Use the cosine law to find angle Q, the smallest angle
10^2 = 17^2 + 21^2 - 2(17)(21)cosQ
cosQ = 630/714 = ....
Q = .....
sinQ = h/17
h = 17(sinQ)