Maths · Distance formula, sections formula, locus and its equation
In the adjoining figure If then
In the adjoining figure \( \boldsymbol{A B}= \) \( 12 c m, C D=8 c m, B D=20 \mathrm{cm} \) \( \angle A B D=\angle A E C=\angle E D C=90^{\circ} . \) If \( B E=x, \) then
- A. \( x \) has two possible values whose difference is 4
- B. \( x \) has two possible values whose sum is 28
- C. \( x \) has only one value and \( x \geq 12 \)
- D. \( x \) cannot be determined with the given information
Step-by-step solution
Place points: B(0,0), A(0,12), D(20,0) since AB=12 and BD=20. Let E(x,0) on BD. With CD=8 and ∠EDC=90°, DC is vertical, so C(20,8). ∠AEC=90° implies AE ⟂ EC. Compute vectors: AE = (-x,12), EC = (20-x,8). Dot product: (-x)(20-x) + 12*8 = 0 ⇒ x² -20x +96 = 0. Solving gives x = 8 or 12. Two values with difference 4.
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