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Question

In the figure 1.89, seg AB $$\parallel$$ seg DC.Using the information given find sum of the the values of x.
185888_d3487dd926cc4e87bf6466a140f04d36.png


Solution

In triangle OAB and OCD,
$$\angle AOB = \angle COD$$ (Vertically opposite angles)
$$\angle BAO = \angle OCD $$ (Alternate angles of parallel lines)
$$\angle ABO = \angle ODC $$ (Alternate angles of parallel lines)
Thus, $$\triangle OAB \sim \triangle OCD$$ (AAA rule)
Hence, $$\frac{OD}{OB} = \frac{OC}{OA} $$ (Ratio of corresponding sides of similar triangles is equal)
$$\frac{3}{x - 3} = \frac{x - 5}{3x - 19}$$
$$9x - 57 = (x - 3) (x - 5)$$
$$9x - 57 = x^2 - 8x + 72$$
$$x^2 - 17x + 72 = 0 $$
$$x^2 - 11x - 6x + 72 = 0 $$
$$(x -8)(x - 9) = 0 $$
$$x = 8, 9$$

Mathematics

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