Answer:
x = -8 + √114 ≈ 2.7 [nearest tenth]
y = 7
Step-by-step explanation:
Given that ∆CAB ≅ ∆CDB, you can use CPCTC [corresponding parts of congruent triangles are congruent] to evaluate the given lengths and angle measures, to show that the corresponding measures are the same for both triangles. Using the triangle sum theorem which states that all angles in a triangle must add up to 180°, you can find ∠ACB.
If ∠CAB = 70°, and ∠ABC = 60°,
∠ACB = 50° because ∠CAB + ∠ABC + ∠ACB = 180°.
∠CAB + ∠ABC + ∠ACB = 180° →
∠ACB = 180° – ∠CAB – ∠ABC
∠ACB = 180° – 70° – 60°
∠ACB = 50°
70° + 60° + 50° = 180°.
Remember, the order of the letters in the angle are very important because it corresponds to the parts of the angle that make up the vertex which is the middle letter, along with the first and last letters which are the endpoints.
Because ∆CAB ≅ ∆CDB,
∠ACB ≅ ∠DCB [CPCTC].
As indicated in the problem, ∠DCB = (x² + 16x)°.
Because ∠DCB ≅ ∠ACB = 50°,
(x² + 16x)° = 50°.
So this can be completed as a square by converting this into vertex form.
x² + 16x = (x + 8)² – 64.
Then, (x + 8)² – 64 = 50.
(x + 8)² – 64 + 64 = 50 + 64 →
(x + 8)² = 114 →
(x + 8) = ±√114 →
x = -8 ± √114 [one more step]
[The measure must be positive so the expression must be added]
x = -8 + √114.
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