Observe Fig. 6.30 and then find ∠P. [Figure: △ABC with AB = 3.8, BC = 6, CA = 33, ∠A=80°, ∠B=60°; and △PQR with PQ = 12, QR = 7.6, RP = 63.]
Get the complete, step-by-step math solution for: "Observe Fig. 6.30 and then find P. [Figure: ABC with AB = 3.8, BC = 6, CA = 3√(3), A = 80°, B = 60°; and PQR with PQ = 12, QR = 7.6, RP = 6√(3).]". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate the third angle of triangle ABC
The sum of angles in any triangle is 180∘. Given ∠A=80∘ and ∠B=60∘ for △ABC, we can find ∠C by subtracting the sum of ∠A and ∠B from 180∘.
Step 2: Substitute and compute ∠C
Substitute the given values of ∠A=80∘ and ∠B=60∘ into the formula for the sum of angles in a triangle to find ∠C. The sum of these two angles is 140∘.
Step 3: Simplify to find ∠C
Subtract 140∘ from 180∘ to calculate the measure of ∠C in △ABC.
Step 4: Compare Side Ratios of Triangles
To determine if the triangles are similar, we compare the ratios of their corresponding sides. We calculate the ratios of AB to RQ, BC to QP, and CA to PR. We observe that all three ratios are equal to 21.
Step 5: Establish Triangle Similarity by SSS criterion
Since the ratios of all corresponding sides are equal, the triangles △ABC and △RQP are similar by the SSS (Side-Side-Side) similarity criterion. Note the correspondence of vertices: A to R, B to Q, C to P.
Step 6: Determine Corresponding Angles
When two triangles are similar, their corresponding angles are equal. From the similarity △ABC∼△RQP, we can establish the correspondence between their angles.
Step 7: Find the measure of Angle P
Given that ∠C=40∘ and knowing that ∠P corresponds to ∠C due to the similarity △ABC∼△RQP, we can conclude that ∠P is also 40∘.