Answer:
21. QR = 10.39 meters
22. m∠P = 60°
23. MN = 9.9 meters
24. NL = 9.9 meters
25. YZ =5√5 meters ≈ 11.18 meters
26. XZ = 10√5 meters ≈ 22.36 meters
Step-by-step explanation:
All three triangles are right triangles
For Δ PQR
PQ = 6, PR, the hypotenuse is 12, find QR
By the Pythagorean theorem
PQ² + QR² = PR²
Plugging in values we get
6² + QR² = 12²
QR² = 12² - 6²
QR² = 144 - 36
QR² = 108
QR = 10.3923 ..
Ans 21. QR = 10.39 to 2 decimal places
To find m∠P
Using triangle ratios for trigonometric functions
cos (m∠P) = side adjacent to ∠P ÷ hypotenuse
= 6/12
= 0.5
m∠P = cos⁻¹(0.5) = 60°
Ans 22: m∠P = 60°
Triangle ΔLMN
This is a right-isosceles triangle since one of the degrees is 45, the other angle must also be 90 - 45 = 45°
LM is the hypotenuse and the other two sides of equal length are the two legs of the triangle
Each equatl side is given by the formula hypotenuse/√2
MN = LM/√2
= 14/√2
= 9.899
≈ 9.9 meters
Ans 23: 9.9 meters
NL = MN = 9.9 meters
Ans 24: NL = 9.9 meters
ΔXYZ
Right triangle with angles 30-60-90
The arm XY = 5√15 m
This is a special right triangle
The sides are in the ratio 1:2:3 with the hypotenuse opposite the right angle and the longer side opposite the larger of the other two angles
Since the larger of the other angles is 60°, the side opposite which is 5√15 length is the longer side
If the shorter arm is of length a meters, then the longer side = a√3 and the hypotenuse is 2a
Since we are given longer arm = 5√15 and shorter arm is YZ we get
YZ x √ 3 = XY
YZ x √ 3 = 5√15
YZ = 5√15 / √ 3
√15 = √3 × √5
5√15 = 5 √5 √3
5√15 / √3 = 5 √5 √3 / √3 = 5√5
YZ = 5√5
Ans 25: YZ = 5√5
26. XZ = 2 YZ = 2 x 5√5 = 10√5
Ans 26: XZ = 10√5
