In trapezoid ABCD, AD∥BC, ∠B=60∘, ∠C=45∘, AD=4, BC=10. Find AB and CD.
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Step-by-Step Solution
Step 1: Draw Perpendiculars and Identify Triangles
To solve this problem, we draw perpendiculars from vertices A and D to the base BC. Let these perpendiculars meet BC at points E and F respectively. This creates two right-angled triangles, △ABE and △DFC, and a rectangle AEFD.
Step 2: Determine Lengths of Segments on BC
Since AEFD is a rectangle, EF=AD=4. We know that BC=10, and BC can be expressed as the sum of BE, EF, and FC. Substituting the known values, we find that the sum of BE and FC is 6.
Step 3: Calculate AE and BE using Trigonometry in Triangle ABE
In the right-angled triangle △ABE, we are given ∠B=60∘. We can use trigonometric ratios to relate AE, BE, and AB. Specifically, tan60∘=BEAE and sin60∘=ABAE.
Step 4: Calculate DF and FC using Trigonometry in Triangle DFC
Since AEFD is a rectangle, AE=DF. In the right-angled triangle △DFC, we have ∠C=45∘. Using the tangent ratio, tan45∘=FCDF. Since tan45∘=1, we get DF=FC.
Step 5: Solve for BE, FC, AE, AB, and CD
From the previous steps, we have AE=BE3 and AE=FC. This implies FC=BE3. We also know BE+FC=6. Substituting FC in the sum equation, we can solve for BE. Once BE is found, we can find FC, AE, AB, and CD using the trigonometric relations.