This exercise involves the construction of a blueprint. To do so, we need to solve for angles using the Cos Rule. See the step by steps procedure below.
<h3>What is a blueprint?</h3>A blueprint is a masterplan that highlights how a structure will be look like when it is completed.
<h3>What is the solution to the above blueprint?</h3>- Let the dimensions of the property be labelled QRST.
- Recall that the distance between the two corners are QR = 210 ft
- Other sides are: 515ft, 232ft, and 542ft going in a clockwise direction.
- Recall that the front left corner of the property is a right angle.
From the above, see attached the first simple quadrilateral QRST (Draft I) that is indicates the above information.
Next Step
We have to derive ∠T, ∠ S, and ∠R.
In order to achieve the above, we must split the quadrilateral into two triangles.
As noted in the image, the quadrilateral is bisected from R to T.
∠R and ∠T are also divided into R1 R2 and T1, T2 respectively.
Next step - Derive ∠T1 and ∠R1 .
To do the above, we must deploy the Pythagoras Theorem. (PT)
PT requires:
TR² = QT² + QR²
TR² = (210)² + (515)²
TR² = 44,100 + 265,225
TR² = 309,325
TR = √309,325
TR = 556.17
Next step, Derive T1
We can derive T1 using the cos rule:
Cos T1 = QR/QT
= 515/556.17
Cos T1 = 0.926
Therefore
T1 = Cos⁻¹(0.926)
T1 = 22.18°
Next we derive R1
R1 = 180° - (90° + 22.18°)
R1 = 67.82°
In ΔTRS, by using Cosine Rule,
Cos T2 = [TS² + QR² - RS²]/2(TS) * (TR)
Cos T2 = [(232)² + (556.16)² - (542)²]/ 2 * (232) x (556.16)
Cost T2 = 69,373.945/258,058
Cost T2 ≈ 0.269
Therefore,
T2 = Cos⁻¹(0.269)
T2 ≈ 74.4°
Applying the same process, we can derive R2
Cos R2 = TR² + RS² - TS²
Cos R2 = [(556.16)² + (542)² - (232)²] / 2 (556.16) * (542)
Cost R2 = 0.911
R2 = Cos ⁻1 (0.911)
R2 = 24.34°
From the above, we can execute the following iterations:
∠S = 180° - (R2 + T2)
= 180 ° - (24.34 + 74.4)
∠C = 81.26°
Likewise:
∠R = R1 + R2
= 67.82 + 24.34
∠B = 92.16°
Lastly:
∠T = T1 + T2
= 22.18 + 74.40
∠T = 96.48°
Learn more about cos rule at;
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Full Question:
A client has just inherited a property. The deed is quite old, but she knows where the front two corners of the property are and that the distance between them is 210 feet. The shape is an irregular quadrilateral and the other three sides have a distance of 515 feet, 232 feet, and 542 feet in a clockwise direction from the front left corner. More importantly, it is known that the front left corner of the property is a right angle.
-Construct a blueprint of the property labeling the measures of all sides and angle of the property.
