927/MR00. problem / Math - Solved Math Problems

MR-927. problem

The sides of a rectangle are 13 and 15 units long. Inscribe a rectangle in the rectangle with one side three times the other. What are the sides of the inscribed rectangle? (The inscribed rectangle has a vertexes on the side of the original rectangle.)

Source: MathReference: Generic Math Textbook - 927.)

Geometry → Two-dimensional geometric shapes → Quadrilaterals

$\frac{x}{y} = 3$

$△ A E H \approx △ B F E$

$\frac{x}{y} = \frac{15 - q}{p} = 3$

$15 - q = 3 \cdot p$

$q = 15 - 3 \cdot p$

$△ A E H \approx △ B F E$

$\frac{13 - p}{q} = \frac{15 - q}{p}$

$p \cdot (13 - p) = q \cdot (15 - q)$

$p \cdot (13 - p) = (15 - 3 p) \cdot [15 - (15 - 3 p)]$

$p \cdot (13 - p) = (15 - 3 p) \cdot (15 - 15 + 3 p)$

$p \cdot (13 - p) = (15 - 3 p) \cdot 3 p$

$13 p - p^{2} = 45 p - 9 p^{2}$

$9 p^{2} - p^{2} + 13 p - 45 p = 0$

$8 p^{2} - 32 \cdot p = 0 | \cdot \frac{1}{8}$

$p^{2} - 4 p = 0$

$p (p - 4) = 0$

$p = 4$

$q = 15 - 3 \cdot p = 15 - 3 \cdot 4 = 15 - 12$

$q = 3$

Both triangles, in addition to being similar, are also right-angled. This allows us to write the Pythagorean theorem:

$y^{2} = p^{2} + q^{2}$

$y^{2} = 4^{2} + 3^{2}$

$y^{2} = 25$

$y = 5$

$x = 3 \cdot y = 3 \cdot 5$

$x = 15$