The coordinates of the vertices of △UVW are U(−2,−1), V(2,−1), and W(2,−8). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree.
A. UV = 4; VW = 7, UW ≈ 7.05
m∠V = 90°, m∠U ≈ 30°, m∠W ≈ 60°
B. UV = 4; VW = 7, UW ≈ 8.06
m∠V = 90°, m∠U ≈ 60°, m∠W ≈ 30°
C. UV = 7; VW = 4, UW ≈ 8.06
m∠U = 90°, m∠V ≈ 60°, m∠W ≈ 30°
D. UV = 7; VW = 4, UW ≈ 7.05
m∠U = 90°, m∠V ≈ 30°, m∠W ≈ 60°
need a bit of help. appreciate it.
Nope. The smallest angle is opposite the smallest side.
gee, did you plot the points? It's clearly a right triangle, with
UW^2 = 4^2+7^2
and V is the right angle.
Now finish it off.
@oobleck so its A
To find the side lengths and angle measures of triangle △UVW, we can use the distance formula to find the lengths of the sides and the distance formula and trigonometric functions to find the angles.
First, let's find the side lengths:
1. UV: Use the distance formula, which is given by d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
So, UV = sqrt((2 - (-2))^2 + (−1 - (-1))^2) = sqrt(4^2 + 0^2) = sqrt(16 + 0) = sqrt(16) = 4.
2. VW: Apply the distance formula: VW = sqrt((2 - 2)^2 + (−8 - (-1))^2) = sqrt(0^2 + (-7)^2) = sqrt(0 + 49) = sqrt(49) = 7.
3. UW: Use the distance formula once again: UW = sqrt((-2 - 2)^2 + (−1 - (-8))^2) = sqrt((-4)^2 + 7^2) = sqrt(16 + 49) = sqrt(65) ≈ 8.06 (rounded to two decimal places).
So, we have UV = 4, VW = 7, and UW ≈ 8.06.
Next, let's find the angle measures:
To find the angle measures, we can use trigonometric functions based on the side lengths.
1. To find ∠U, we can use the cosine rule: cos(U) = (VW^2 + UW^2 - UV^2) / (2 * VW * UW).
Plugging in the values: cos(U) = (7^2 + 8.06^2 - 4^2) / (2 * 7 * 8.06) ≈ 0.154.
Taking the inverse cosine (cos^-1) of 0.154, we find that ∠U ≈ 81.79° (rounded to the nearest degree).
2. To find ∠V, we already know that it is a right angle (∠V = 90°).
3. To find ∠W, we can use the sine rule: sin(W) = (VW * sin(U)) / UW.
Plugging in the values: sin(W) = (7 * sin(81.79°)) / 8.06 ≈ 0.869.
Taking the inverse sine (sin^-1) of 0.869, we find that ∠W ≈ 59.55° (rounded to the nearest degree).
So, the correct answer is D.
UV = 7; VW = 4, UW ≈ 7.05
m∠U = 90°, m∠V ≈ 30°, m∠W ≈ 60°.